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Book_ t \ \ . . 

Copyright N°__ 


COPYRIGHT DEPOSIT 













Alternating Current 


MOTORS 


BY 


A. S. MCALLISTER, Ph. D. 

ii 


SECOND EDITION REVISED AND ENLARGED 


) 

) 

) 

> 

> t 



NEW YORK 

MCGRAW PUBLISHING COMPANY 


1907 








\ LIBRARY of CONGRESS 

I 

Two Gnoles Received 

OCT 2 190/ 

Copyright Entry 
£ c. t z 6 / ? * 7 

CLASS A XXC„ No, 
/ ? C 
COPY B. 


Copyrighted, 1906, 1907, 
by the 

McGraw Publishing Company, 
New York. 


% 

f c 








PREFACE TO THE SECOND EDITION. 


A comparison of the present edition with the first edition of 
this book will show that the magnetic field in the single-phase 
induction motor is treated in more detail, a complete discussion 
of the operating characteristics of the recently developed “ motor- 
converter ” has been given in Chap. IV, on Frequency Converters, 
with which Chap. V has been combined and a Chapter on the 
Leakage Reactance of Induction Motors has been added in the 
form of an Appendix. In accordance with suggestions received 
from numerous competent readers, including many instructors 
in electrical engineering now using the book in their classes, who 
kindly responded to the author’s request for criticisms, a few 
minor changes have been made at other places throughout the 
book, in order to render the treatments more readily understood. 

The book has been written and revised exclusively for the 
reader,—and the particular reader for whom the book is in¬ 
tended is assumed to possess a fair general idea of electro¬ 
magnetic phenomena and wishes to acquire specific information 
concerning the various types of alternating-current motors. 
In selecting the methods of presentation, choice has been made 
of the treatments w T hich serve to impart the maximum of informa¬ 
tion with the minimum of confusion. As one reviewer has aptly 
stated, “ The general method of treating all motors is first to 
outline the theory roughly, leaving out the less important 
factors, and then to go through again putting in the details.” 
For example, in Chap. XII the physical phenomena involved in 
the operation of single-phase commutator motors are treated in 
the simplest possible manner without any reference to the me¬ 
chanical dimensions or the electrical constants; in Chaps. XIII 
and XIV, however, these motors are discussed in detail and the 
exact relation between the several parts of each machine is 
treated both algebraically and graphically. The phenomena of 
induction motors are dealt with separately in Chap. II, Chap. 
VI, Chap. VIII and Chap. IX. To one thoroughly familiar 

iii 



IV 


PREFACE . 


with the characteristic performance of induction motors, it 
might seem that these four chapters contain a considerable 
amount of repetition. The reader who is using the book as the 
source of his first information concerning the motors, however, 
will find that in Chap. II the entire treatment is concentrated on 
the secondary circuit, the modifying influences of the primary 
quantities being temporarily ignored; the secondary circuit is 
considered as it actually exists—a combination of a constant 
resistance and a variable reactance. In Chap. VI it is shown that, 
in its effect upon the primary quantities, the secondary circuit 
may be considered as composed of a constant reactance and a 
variable resistance , and the primary and secondary quantities 
are combined and treated graphically (see Fig. 35) by a method 
■whose most prominent characteristic is its simplicity. Chap. 
VIII gives a rigorous discussion of the operation of induction 
motors, shows how to construct an accurate current locus of the 
machine (Fig. 53), and explains the use of circle diagrams for 
both polyphase and single-phase motors, in which all errors are 
practically neutralized (Figs, 54 and 56). Chapter IX contains 
an analysis of the magnetic field in induction motors, which is 
believed to be absolutely accurate within the limits specified. 
Certain portions of this chapter, especially that dealing with the 
single-phase motor, cannot be considered “ simple,” because the 
electromagnetic phenomena involved are extremely complex. 
The ordinary method of treating the facts covered in these four 
chapters would be to present the substance of Chap. IX first, 
and then to follow with Chaps. VIII, VI and II, in the order here 
given. The present writer believes that, viewing the subject 
from the standpoint of the reader for whom the book is intended, 
this method is essentially wrong, because it tacitly assumes that 
the reader possesses initially just that amount of information 
w r hich the treatment when concluded is expected to impart. 

To each of those who have added to the value of the present 
edition as a book for the reader, by reporting observations of 
the first edition from the viewpoint of the reader, the author 
desires to express his sincere thanks. 



PREFACE TO FIRST EDITION. 


In the preparation of the explanatory matter contained in 
the present volume, the writer has had constantly in mind the 
difficulties which he encountered in endeavoring to ascertain 
the causes for the various effects which were described in the 
available electrical literature in connection with the operation 
of the several types of alternating-current motors. An exper¬ 
ience of several years in instructing engineering students in 
alternating-current phenomena has served to convince the 
writer that the above mentioned difficulties are due primarily 
to the methods employed in presenting the facts to minds un¬ 
familiar therewith. It is believed that a large part of the 
difficulty may be attributed to an inconsistent use of technical 
terms, such as the word “ field,” meaning at one time mag¬ 
netism, at another iron, and yet at another copper. Many 
writers use the term “ power ” without discrimination for 
quantities measurable in watts or in watt-hours. It is surpris¬ 
ing how frequently even well-informed writers use the term 
“ current ” when “ e.m.f.” is intended. One often sees the state¬ 
ment “ a current is induced,” when, as a matter of fact, the 
circuit in which the current is supposed to flow is entirely open. 
Much of the difficulty is due to the excessive use of mathematical 
equations as an end rather than a means. It is believed that 
mathematics should be employed merely as a short-hand method 
of stating facts. Beyond any doubt much confusion is created 
in the mind of an engineering student when an attempt is made 
to express certain assumed relations in the form of mathematical 
equations, and then to transform the equations to obtain a 
result which he is asked to believe expresses physical facts, 
merely because the equations, considered mathematically, have 
been properly transformed. 

In view of the facts stated above, and on account of the 
belief that, for the purpose of imparting information, simplicity 
is the criterion of worth, an attempt has been made to deal 
directly with the electromagnetic phenomena of alternating- 
current motors in the simplest possible manner, in so far as such 
method is consistent with accuracy. Wherever mathematical 



Vl 


PREFACE. 


equations are employed, the assumption upon which they are 
based are definitely stated, and the inaccuracies in the assump¬ 
tions and limitations in the equations are carefully noted, while 
the results obtained from the transformed equations are inter¬ 
preted with due regard to the inaccuracies and the limitations. 
Moreover, the significance of each transformation is carefully 
explained, so that the reader may be constantly reminded of 
the fact that he is dealing directly with electromagnetic phe¬ 
nomena and only indirectly with mathematics. 

It has been assumed that the reader is familiar with the funda¬ 
mental facts of electricity and magnetism, and that he has some 
knowledge of the lower branches of mathematics. No attempt 
has been made to explain the manner in which alternating 
electromotive forces and currents may be represented in value 
and time-phase position by straight lines, because a knowledge 
of such representation can be presupposed. Much attention is 
given to graphical diagrams and to examination of the facts upon 
which they are based. Where necessary the errors involved in the 
assumptions are pointed out, and the magnitude of their effect on 
the final results are discussed. In developing the graphical dia¬ 
gram of the induction motor, the proof that the current locus is 
(approximately) a circle has been based on the relation between 
the equivalent electric circuits considered as certain intercon¬ 
nected resistances and reactances, rather than on the resolution 
of the magnetic flux into certain components. The latter 
method is the one most commonly employed. It is believed, 
however, that the former method possesses peculiar advantages 
in permitting the reader to follow the development step by step 
without losing sight of the involved electromagnetic relations, 
and in allowing the inaccuracies in the assumptions to be defi¬ 
nitely determined. 

The major portion of the volume has appeared as articles in 
various publications, notably the Electrical World, the American 
Electrician and the Sibley Journal of Engineering. The writer’s 
thanks are due to the editors of these publications for permis¬ 
sion to incorporate the articles in this book. The writer 
wishes to take this opportunity to express his appreciation of 
the encouragement continuously received from his friend and 
former colleague, Dr. Frederick Bedell. 



COISTTETSTTS 


CHAPTER I. 

Single-Phase and Polyphase Circuits... 1 

Economy of Conducting Material. 1 

Three-Phase Power Measurements. 3 

Measuring Three-Phase Power with One Wattmeter. 6 

Wattmeters on Unbalanced Loads. 9 

Equivalent Single-Phase Currents. 13 

Equivalent Single-Phase Resistance. 14 

Advantages of Employing Equivalent Single-Phase Quantities.. 15 

CHAPTER II. 

Outline of Induction Motor Phenomena . 16 

Methods of Treatment. 16 

Production of Revolving Field. 17 

Simple Analytical Equations. 19 

Starting Devices for Induction Motors. 22 

Concatenation Control. 23 

CHAPTER III. 

Observed Performance of Induction Motor. 25 

Test with One Voltmeter and One Wattmeter. 25 

Measurement of Slip. 26 

Determination of Torque. 27 

Measurement of Secondary Resistance. 28 

Determination of Secondary Current. 28 

Calculation of Primary Power Factor. 29 

Calculation of Primary Current. 30 

CHAPTER IV. 

Induction Motors as Frequency Converters . 33 

Field of Application. 33 

Characteristic Performance. 34 

Capacity of Frequency Converters. 35 

Motor Converters. 36 

Division of Power between the Motor and the Converter. 37 

Advantages of the Motor Converter. 38 

Excitation of Motor Converters. 39 

vii 
































CONTENTS. 


viii 

Phases of the Motor Converter. 40 

Uses of the Motor Converter. 40 

Utilization of Frequency Converter Properties. 41 

Direct Current in Secondary Coils. . . 42 

Alternating Current in Secondary Coils. 42 

Action with Stationary Rotor. 44 

Action with Rotor at Synchronous Speed. 46 

CHAPTER V. 

The Single-Phase Induction Motor . 48 

Outline of Characteristic Features. 48 

Production of Quadrature Magnetism. 49 

Production of Revolving Field. 51 

Elliptical Revolving Field. 52 

Starting Torque of the Single-Phase Motor. 53 

Use of “ Shading Coils ”. 54 

Use of Commutator on the Rotor. 56 

Polyphase Induction Motors Used as Single-Phase Machines_ 58 

CHAPTER VI. 

Graphical Treatment of Induction Motor Phenomena . 63 

Advantage of Graphical Methods. 63 

Effect of Inserting Resistance in the Secondary. 63 

Primary and Secondary Current Locus. 65 

Test Results. 66 

Equation of the Current Locus. 70 

Errors in Assuming a Circular Arc. 71 

Effect of Design on Leakage Reactance. 73 

CHAPTER VII. 

Induction Motors as Asynchronous Generators .'. 74 

Operation Below Synchronism. 74 

Operation Above Synchronism. 75 

Vector Diagram of Currents. 75 

Performance Characteristics. 79 

Parallel Operation of Asynchronous Generators. 80 

Excitation of Asynchronous Generators. 81 

Condensance as a Source of Exciting Current. 83 

Condensers in Alternating Current Circuits. 84 

Excitation Characteristics of Asynchronous Generators. 86 

Load Characteristics of Asynchronous Generators. 90 

Commutator Excitation of Asynchronous Generators. 92 

CHAPTER VIII. 

Transformer Features of the Induction Motor . 95 

Electric and Magnetic Circuits. 95 

Equivalent Electric Circuits. 98 








































CONTENTS. 


IX 


Modified Electric Circuits. 99 

Circle Diagram of Currents. 100 

Internal Voltage Diagram of the Induction Motor. 105 

Corrected Current Locus of the Induction Motor. 107 

Complete Performance Diagram of the Polyphase Induction 

Motor. 109 

Comparison of Single-Phase and Polyphase Motors. 112 

Electric Circuits of the Single-Phase Induction Motor. 114 

Complete Performance Diagram of the Single-Phase Induction 

Motor. 115 

Speed and Torque of the Single-Phase Induction Motor. 117 

Capacities of Single-Phase and Polyphase Motors. 119 

CHAPTER IX. 

Magnetic Field in Induction Motors . 121 

Polyphase Motors. 121 

Magnetic Distribution with Open Secondary. 124 

Magnetic Distribution with Closed Secondary. 127 

Determination of Core Flux. 130 

Effect of Core Flux of Using Distributed Winding. 131 

Effect on Capacity of Varying the Grouping of Coils. 133 

Exciting Watts in Induction Motors. 135 

Magnetic Field in the Single-Phase Induction Motor. 138 

Production of Speed-Field Current. 139 

Transformer Features of the Single-Phase Induction Motor. 142 

Analysis of the Rotor Electromotive Forces. 143 

Secondary Currents in the Single-Phase Motor.143b 

Graphical Representation of Secondary Quantities. 146 

CHAPTER X. 

Synchronous Motors and Converters . 149 

Synchronous Commutating Machines. 149 

Synchronous Motors and Generators. 151 

Synchronous Converters, Unity Power-Factor. 154 

Distribution of Heat Loss in Armature Coils. 159 

Synchronous Converters,. Fractional Power-Factor. 160 

Double Current Machines. 161 

Relative Capacities of Synchronous Machines of Various Phases. 163 

Characteristic Performance of Synchronous Converters. '... . 165 

Excitation of Synchronous Machines. 166 

Hunting of Synchronous Machines. 167 

Starting of Synchronous Converters. 168 

Compounding of Synchronous Converters. 169 

Inverted Converters.. .. 170 

Predetermination of Performance of Synchronous Converters... 171 

Six-Phase Converters. 174 

Six-Phase Transformation. 175 

Relative Advantages of Delta and Star-Connected Primaries.... 180 









































CONTENTS. 




CHAPTER XI. 

Electromagnetic Torque . .. 181 

Commutator Motors. 181 

Equality of Torques for Uniform Reluctance. 181 

Inequality of Torques for Non-Uniform Reluctance. 183 

Determination of Torque by Calculation of the Output. 184 

Measurement of Torque by the Loading-Back Method. 185 

Elimination of Errors. 187 


CHAPTER XII. 

Simplified Treatment of Single-Phase Commutator Motors _ 189 


The Repulsion Motor. 184 

Electric and Magnetic Circuits of Ideal Motor. 190 

Production of Rotor Torque. 192 

Graphical Diagram of Repulsion Motor._. 193 

Calculated Performance of Ideal Repulsion Motor... 196 


CHAPTER XIII. 

Motors of the Repulsion Type Treated Both Graphically and 


Algebraically . 199 

Electromotive Forces Produced in an Alternating. Field. 199 

The Simple Repulsion Motor. 201 

Effect of Speed on the Stator Electromotive Forces. 203 

Fundamental Equations of the Repulsion Motor. 204 

Vector Diagram of Ideal Repulsion Motor. 206 

Corrections for Resistance and Local Leakage Reactance. 209 

Brush Short-Circuiting Effect. 212 

Observed Performance of Repulsion Motor. 213 

Compensated Repulsion Motor. 214 

Apparent Impedance of Motor Circuits. 217 

Fundamental Equations of the Compensated Repulsion Motor..218 

Vector Diagram of Compensated Repulsion Motor. 221 

Calculated Performance of Compensated Repulsion Motor. 223 

Observed Performance of Compensated Repulsion Motor. 225 

Corrections for Resistance and Local Leakage Reactance. 227 

Brush Short-Circuiting Effect.:. 228 

CHAPTER XIV. 

Motors of the Series Type Treated Both Graphically and Al¬ 
gebraically . 232 

The Plain Series Motor. 232 

Fundamental Equations of Series Motor with Uniform Air-Gap 

Reluctance. 233 

Fundamental Equations for Motor with Non-Uniform Reluct¬ 
ance . 237 

Inductively Compensated Series Motor. 241 

Conductively Compensated Series Motor. 242 




































CONTENTS. 


xi 


Complete Performance Equations of Compensated Motors. 243 

Vector Diagram of Compensated Series Motor. 244 

Induction Series Motor. 245 

Fundamental Equations of Induction Series Motors. 247 

Corrections for Resistance and Local Leakage Reactance. 252 

Vector Diagram of Induction Series Motor. 253 

Generator Action of Induction Series Motor. 254 

Brush Short-Circuiting Effect. 255 

Hysteretic Angle of Time-Phase Displacement. 257 

Power Factor of Commutator Motors. 260 

Resistance in Shunt with Field Winding. 260 

Loss Due to Use of Shunted Resistance. 263 

CHAPTER XV. 

Prevention of Sparking in Single-Phase Commutator Motors.. 266 

Transformer Action with Stationary Rotor. 266 

Interlaced Armature Windings. 267 

Use of Series Resistance. 267 

Power Lost in Resistance Leads. 268 

Internal Resistance Leads. 269 

External Resistance Leads with Two Commutators. 269 

External Resistance Leads with One Commutator. 270 

APPENDIX. 

The Leakage Reactance of Induction Motors . 273 

The Leakage Coefficient. 273 

Exciting Current by Combined Magnetomotive Force Method. . 275 

Exciting Current by Single-Phase Method. 276 

Leakage Reactance Equations. 277 

Main Dispersion Factor. 2/9 

Zigzag Dispersion Factor. 281 

Reactance of Coil-Wound Secondaries and Squirrel-Cage Motors . 282 

Reactance of Fractional Pitch Windings . 283 

Equivalent Single-Phase Reactance and Starting Current. 283 

Calculation of Synchronous No-Load Currents. 284 

















































































































































































































































CHAPTER I. 


SINGLE-PHASE AND POLYPHASE CIRCUITS. 

Economy of Conducting Material. 

Although of all possible transmission systems the single-phase 
requires the least number of conductors and, on the basis of 
equality of maximum e.m.f. to the neutral point, the single¬ 
phase is the equal with reference to the cost of conductors of 
any polyphase system, practically all of the long-distance trans¬ 
mission circuits are of the polyphase type, chiefly because of the 
facts that polyphase generators are less expensive in construc¬ 
tion and more economical in operation than single-phase ma¬ 
chines, and, for the purpose of power distribution, polyphase 
motors and synchronous converters are superior to single-phase 
machines. Of all polyphase circuits operated with a given 
maximum measurable e.m.f. between lines, the three-phase sys¬ 
tem is the most economical with reference to the cost of con¬ 
ducting material; which accounts for the fact that, with very 
few exceptions, all transmission circuits are of the three-phase 
type. 

That all symmetrical transmission systems show the same 
economy of conducting material, irrespective of the number of 
phases, when compared on the basis of equality of maximum 
e.m.f. to the neutral point, can be proved as follows: Let P = 
the number of phases (and conducting wires); R = the resist¬ 
ance of the total mass of conducting material, all wires being 
considered in parallel; E = the e.m.f. to the neutral point, 
and W — the power transmitted. Then P R = resistance per 

wire; — = power per phase, and ■ - = current per wire. 

Representing the current per wire by /, the loss in transmission 
is, 

/ W V W 2 

PPPR-ffi) P 2 R = R. 

which is obviously independent of the number of phases. 

1 



2 


ALTERNATING CURRENT MOTORS. 


For all systems in which the measurable maximum e.m.f. is 
twice the e.m.f. from a single conductor to the neutral point, 
the cost of conducting material for transmitting at a given 
loss is the same. In this class fall the two-wire (single-phase), 
four-wire (two-phase), six-wire (six-phase), and other systems in 
which the number of phases is an even one. An inspection of 
Fig. 1 will show that in the three-phase system the measurable 
e.m.f. is only 

2E cos i (“p”) = V3 X E, 


or 1.732 times that from one conductor to the neutral point. 
On the basis of equality of measurable e.m.f., therefore, the 


three-phase system requires 


m- 


as much conducting 



Fig. 1 . —E.m.f.’s of Fig. 2. —E.m.f.’s of 
Three-phase Circuits. Six-phase Circuits. 


material as any system in which the number of phases, P, is 
even, since the conducting material varies inversely as the 
square of the e.m.f. for any given system. The calculations re¬ 
corded in Table I show that a higher conductor efficacy is 
obtained in each case when P is odd than when it is even, and 
that the smaller the value of P the higher the efficacy will be. 
Since P, when odd, cannot be less than three, the three-phase 
is of all systems the most economical. 

While as regards the desirability of obtaining economy in 
cost of conducting material, the problem of determining the 
proper circuits for the distribution of electrical energy at the 
end of a transmission line is quite similar to that connected 
with the transmission circuits, factors other than economv 
enter into this portion of the problem, which must be carefully 
considered before any attempt is made at a definite solution. 




SINGLE-PHASE AND POLYPHASE CIRCUITS. 


3 


TABLE I.— Relative conducting 

material 

required for different 

tranrr.isssion 

systems 

(/) 

w 

On basis of 
minimum e.m.f. 

On basis of max¬ 
imum measurable e.m.f. 

Number of pi 
(wii es) 

a 

Conducting 

material 


3 

a 

M 

4 

a 

Conducting 

material 





* 



P 

IB' 

CQ 

<N 

N 

G 1 

1*3! 



00 jQs 

10 

o 

o 

N 

3? 

a^ 

2 

2.0000 

1 .000 


2.0000 

• . • • 

1 .000 

3 

1 .7320 

.750 


> • • • • 

1.7320 

.750 

4 

1.4142 

.500 


2.0000 

.... 

1.000 

5 

1.1756 

.346 


.... 

1.9022 

.905 

6 

1.0000 

.250 


2.0000 

.... 

1 .000 

7 

.8678 

.188 


.... 

1.9500 

.950 

8 

.7654 

.146 


2.0000 

.... 

1 .000 

9 

.6840 

.117 


.... 

1.9696 

.969 


Three-Phase Power Measurements. 

In measuring the power in a three-phase circuit the most 
convenient method is one involving the use of two wattmeters 
whose current coils are placed in any two of the three phase 
leads, and whose pressure coils are connected, respectively, 
between these two leads and the third lead. A simple semi- 
graphical proof of the correctness of the two-wattmeter method 
of measuring the power in a three-phase circuit under any 
condition of service is given below. 

In Fig. 3, let the sides of the triangle ABC represent the 
relative values and phase positions of the three e.m.fs. of an 
unsymmetrical three-phase system. Assume the receiver to be 
delta connected, and let I AB be the current in coil AB } and 0 AB 
be the angle of lag of this current with respect to the e.m.f. 
of the coil. Similarly, let I BC and I AC represent the value and 
phase position of the currents in coils B C and A C. No restric¬ 
tion is made as to the values of currents, e.m.fs. or as to the 
several lag angles. 

















4 


ALTERNATING CURRENT MOTORS. 


Let a wattmeter be connected with its current coil in lead at 
A, and its e.m.f. coil across between this lead and lead C. Also a 
wattmeter at B, with pressure coil between B and C. Each 



Fig. 3.— Phase Relation of e.m.f. ’s and currents. 

wattmeter will show a deflection, which may be represented by 
W = E I Cos 0, where I is the amperes in the current coil, E is 
the volts across the e.m.f. coil, and 0 is the angle between this 
e.m.f. and the current I. 



Fig. 4.—Measurement of power in coils A C and C B. 

For sake of simplicity in explanation, assume, in the first 
place, the current flowing in coil A B to be absent while measure¬ 
ments are made upon the watts supplied to the other coils, as 
indicated by Fig. 4. Evidently the sum of the readings of the 













SINGLE-PHASE AND POLYPHASE CIRCUITS. 


5 


meters as connected gives the watts in the two remaining coils, 
since each meter is connected as though measuring power in a 
single-phase circuit. Now assume, in the second place, the 
current to flow in coil A B alone while the currents in the other 
two coils are absent, as shown by Fig. 5. According to proof 
given below, the wattmeters as connected now register as their 
sum the true watts supplied to coil A B. When all three cur¬ 
rents flow simultaneously, each wattmeter will show a de¬ 
flection equal to the sum of its two previous readings, since its 
e.m.f. coil has undergone no change in connection and the two 
currents causing the former deflections are now superposed, and 
the true power transferred will be properly recorded by the 
two meters. 



Fig. 5.—Measuring power in coil A B. 

Two wattmeters having their current coils in series with a 
given single-phase load, and one terminal of the e.m.f. coil of 
each meter connected to the opposite leads of the ciicuit supply¬ 
ing power to the load and the other two free terminals con¬ 
nected together and placed at any point of any relative poten¬ 
tial compared with that of the load, as depicted in Fig. 5, will 
give the true value of power transmitted. 

& In Fig. 6, let E A b be the e.m.f. across the load, I A b be the 
load current and 6 a b be the angle between I AB and E AB . Evi¬ 
dently the watts transmitted are 

W A B = E A B I ab COS 0 A b • 











6 


ALTERNATING CURRENT MOTORS. 


Now assume a wattmeter connected at A to C. Its reading 
will be 

Wac = Eac Iab cos Oac • 

A wattmeter at B to C will read 

W bc = Ebc Iab cos Qbc- 

From Fig. G, 

Eab cos Oab = A F. 

Eac cos Oac = A E. 

Ebc cos Qbc = B D = E F. 
and since AF = AE + EF, 

Iab ( Eac cos Qac + Ebc cos Qbc) = Iab Eab cos Oab 
or Wac + Wbc = Wab 

and this value is independent of the position of the point C. 



Fig. 6 .—Graphical proof of Fig. 5. 


In consequence of this fact, P — 1 wattmeters may be used to 
determine the true power in any P —phase system, however un- 
symmetrical may be the phase relations, provided the free ter¬ 
minals of the e.m.f. coil of each meter be connected to that 
lead in which no wattmeter is placed. 

Measuring Three-Phase Power with One Wattmeter. 

A single wattmeter method which may be used for deter¬ 
mining the angle of lag in balanced three-phase circuits is out¬ 
lined below. It is believed that this method is not as generally 
well known as its simplicity and comparative freedom from 
errors justify. 

If a wattmeter, connected as indicated by the diagram of 






SINGLE-PHASE AND POLYPHASE CIRCUITS. 


7 


Fig. 7, with its current coil in one lead of a three-phase circuit, 
be read with its e.m.f. coil across between this lead and first one 
and then the other of the remaining two leads, the two values 
thus obtained may be used to determine the angle of lag of the 
current by means of the following relation: 

W t - W 2 
tan<j& -s/z Wi + w 2 

where cj) is the angle of lag, and W v W 2 are two readings of the 

wattmeter, as indicated above. 

When cf> is greater than 60 degs. one reading will be negative 
so that the difference of readings will be greater than their sum. 



The proof of this relation is as follows: 

L e t I = current in lead A X , 

E= e.m.f. of A B and of A C, 

t ^ ien 1 Wi = I E cos ($ — 30) 

W 2 = I E cos (<f> + 30) 

and since ^ (a ±/5) = cos a cos /? T sin a sin (I 

__W 2 = I E [cos ( <f> —30) — cos (<j> + 30) ] 
= 2 I E sin 30 sin <f> = / E sin 4> 

an W l + W., = I E [cos (cf> — 30°) -\-cos (</> + 30) ] 
= 21 E cos 30° cos$ = Vs I E cos <f> 

hence 


^- 2 = -i- tan 6, as above. 

W L + W 2 V 3 













8 


ALTERNATING CURRENT MOTORS. 


If the e.m.f. of A B be not equal to the e.m.f. of A C, the 
reading of the wattmeter in either position may be corrected to 
such a value as would have been obtained had the two voltages 
been equal, in which case the above relation will hold true. 

Since any proportional error in the calibration scale of the 
wattmeter affects equally both the sum and the difference of 
the readings, and hence does not alter the ratio of the two, it 
follows that a wattmeter having a proportional scale error of 



any value whatever may be used to obtain the correct value 
of lag angle, and that any instrument of the dynamometer type, 
whether calibrated or not, may be used in place of the wattmeter. 

The lag angle thus obtained is the true angle between the 
current in lead A X and the mean voltage between A B and A C. 

If the circuits are symmetrically loaded the sum of two 
correct wattmeter readings will equal the real power, while 
from the difference of the two readings may be obtained the 
“ quadrature ” watts. 




























SINGLE-PHASE AND POLYPHASE CIRCUITS. 


9 


Wattmeters on Unbalanced Loads. 

It is to be noted especially that the above method must be used 
with care, because an unbalance of load may lead to incorrect 
interpretation of the results. An exaggerated case of unbalance, 
selected so as to emphasize the facts just stated, is shown 
below. In order that all disturbing influences, other than the 
unbalance of the load alone, may be eliminated from the prob¬ 
lem, there has been assumed a non-inductive load supplied from 
a three-phase circuit having equal e.m.fs. between the leads. 

Fig. 8 represents the circuits and load, and shows the con- 



Fig. 9.—Vector Diagram of Currents and Electromotive Forces. 

nection of instruments for determining the power factor. As 
seen, an e.m.f. of 100 volts between leads and a delta-connected 
non-inductive load of 10, 20 and 30 amperes, respectively, per 
phase have been chosen. The true power is evidently 6,000 
watts, while the power factor per phase is unity. 

Referring to Fig. 9, which represents the value and position 
of the current per phase of the load indicated by Fig. 8, the 
value of current registered upon an ammeter at C and its rela¬ 
tive phase position may be ascertained by making use of the 
geometrical figure c d f 6. From the construction it is seen that 
the current at c is represented in value and phase by the line c j. 








10 


ALTERNATING CURRENT MOTORS. 


In the triangle c e /, the side cf is equal to 

(c e 2 + e f 2 + 2 c e . e f cos 120°)I 

or is equal to 

(ce 2 -\-ceef-\-e f 2 )% = \/1900 = 43.60 and 
sin<i> c = sin 120° = .866 NN = .3971 

<j>c = 23° 24'. 

Similarly the current at a is 

(10 2 +10x 20 + 20 2 )* = \/700 = 26.45, and 

• A. QCC 10 - 00 0979 

sin (b a = . 866 ^ = .32/2 

^ 26.4o 

4> b = 19° 6'. 

Current at 6 is 

(10 2 + 10 x 30 + 30 2 )! = \/h300 = 36.05 and 

sin chb = .866 = .2402 

36.05 

= 13° 54'. 

It is convenient to adopt some method of designating at once 
each wattmeter and its connection in the circuit. Place, there¬ 
fore, as subscript to the letter W, which is to represent the 
reading of each meter, the letters showing the points between 
which the voltage coil is connected, and place first that letter 
corresponding to the lead in which is the current coil of the 
wattmeter. Thus, W a b refers to wattmeter having its current 
coil in lead a and its voltage coil connected across between this 
lead and lead b. 

Wattmeter W a c will record I a E a c Cos cp a c, or 
Wac = 26.45X 100Xcos 19° 6' = 2,500. 

W a b = 26.45 X 100 Xcos 40° 54' = 2,000. 

W be = 36.05 X lOOXcos 13° 54' = 3,500. 

Wba = 36.05X 100X cos 46° 06' = 2,500. 

W c b = 43.60X lOOXcos 33° 24' = 4,000. 

Wca = 43.60X 100Xc<?s 36° 36' = 3,500. 

It is seen at once that the true value of watts is recorded in 
each case by the sum of the readings of any two wattmeters 






SINGLE-PHASE AND POLYPHASE CIRCUITS. 


11 


with their current coils in separate leads and their free pressure 
terminals connected to the third lead, thus (W ac + W b c) = 
(W ab + Wcb ) = (W b a + W c a) = 6,000, but that the true 
watts may not be indicated by one wattmeter which has its 
pressure coil free terminal transferred from first one and then 
the other remaining lead, thus (W ac + W a b) — 4,500, (W be 
+ Wba) = 6,000, (Web + W c a) = 7,500. 

It was shown above that the angle of lag and the power 
factor may be determined by the ratio of the readings of two 
wattmeters. That such a method does not give accurate re¬ 
sults with unbalanced loads was mentioned also. For purpose 
of comparison, however, results determined by this method 
are here recorded. Letting 0 represent the general angle of lag, 


_ W b c - W a c ^ 1000 
tan d ~ Vd w bc+ W ac V3 6000 


.2886 


6 

cos 6 
tan 0 
Q 

cos 6 
tan 6 

6 


16° 6' 

.961 

,_Wcb — Wab 
V 3 


W cb+W ab 

30° O' 

.866 

W c a — Wba 
v ' 3 W ca+ Wba 

16° 6' 


= v 7 3 


2000 

6000 


= .5772 


1000 

x/3 6000 


= .2886 


cos 6 = .961 


Since the load has been so selected as to be strictly non- 
inductive, it is evident that the lag angle indicated does not 
exist, and that the power factor obtained by this method is in 


error. 

It is to be noted in this connection, however, that the angle 
of lag obtained by the same formula used above, but sub¬ 
stituting the ratio of readings of one wattmeter when its pressure 
coil is transferred between the two leads, as mentioned above, 
has, in fact, a physical significance, as here shown: 


tan 6 a — \/3 


_W a c — W a b 


W~ac+Wnb 


= V 7 3 


500 

4500 


.1925 


6 


a 


= 10° 45 










12 


ALTERNATING CURRENT MOTORS. 


An inspection of Fig. 2 will reveal the fact that this is the 
angle between the current at a and the mean voltage between 
ab and ac, since 10° 54' = 30° — (19° + 6')- Similarly, 


. ^Wbc-Wba 1000 

tan 6 b - V 3 w b c + w b a - V 3 6000 - 


.2886 


and again, 


0 b = 16°. 6' = 30° - (13°. 54') 


Q W c b — W c a 500 e _ 

tan6c = v/3 W cb+ W ca~ 7500 “ ' 1155 

(J c = 6° 36' = 30° - (23°. 24') 

A popular formula for determining the power factor of a three- 
phase load is 

W 


P. F. = 


VS EI 


where I is the current per lead wire. Substituting the values 
found above for I, the power factor is 


P. F. 


c 


6000 

173.2X43.60 


.795 


P.F.h = 


6000 


173.2X36.05 


= .959 


P.F. a 


6000 


173.2X26.45 


= 1.309 


P. F . 


.795+ .959+1.309 
3 


1.021 


Using as a value for I the mean current per lead wire, 


P. F. 


6000 

173.2X35.37 


.980 


Several of the methods used above are obviously in great 
error, and their use would never be sanctioned in a careful test. 
Few objections, however, could be raised against the last two 
methods of averages, though neither gives the true result. 

In the determination of the power factor as the ratio of true 
to apparent power, the question arises as to what constitutes 
the apparent power, and the discrepancies in results are due 
to the various answers which may be given to this question. 












SINGLE-PHASE AND POLYPHASE CIRCUITS. 


13 


While doubt must ever exist as to the value to be assigned 
to the apparent power in a three-phase system operating on an 
unbalanced load, the method in common use for determining 
the true power is correct for any condition of load, proportion 
of e.m.fs. or relation of power factor of currents, though the 
methods of proof of this fact, which are based on assumptions 
of equal currents, equal power factors, or equal e.m.fs. per 
phase, are evidently open to many objections. 

Equivalent Single-Phase Currents. 

Though some advantages may be claimed for the method of 
dividing the amount of power supplied to a polyphase motor by 
the number of phases and then treating the machine as that 
number of single-phase motors, greater simplicity is introduced 
into the calculation if equivalent single-phase qualities be de¬ 
termined for the polyphase circuit. In accordance with this 
plan the total number of watts supplied to the circuit is used 
in computations without alteration, the measured e.m.f. of the 
polyphase circuit is considered the equivalent single-phase 
e.m.f., while the equivalent effective current is understood to 
mean that value of current which must flow at the same power- 
factor in a single-phase circuit at the same voltage to transmit 
the same power. 

It is evident that in a two-phase circuit the sum of the cur¬ 
rents of the separate phases is the equivalent single-phase 
current. This quantity will hereafter be referred to as the 
“ total current,” or as simply the “ current ” of the two-phase 
circuit. Since the total power transmitted in a three-phase 
system is expressed by 

W = n /3 I E cos 

\/ 3 1 is the equivalent effective current. In the equation 
above, / is the current per wire. For a delta-connected receiver 
the current per phase is equal to / -f- \/3 or the sum for the three 
phases is 3 /-h\/3 = a/3 T The quantity ^3 I has, therefore, 
a physical significance as the total current in a delta-connected 
receiver, though in a star-connected machine such quantity 
exists only mathematically. However, for the sake of com¬ 
bined generality of treatment and brevity of discussion, y/%1 
will hereafter be spoken of as the “ total current,” or the “ cur¬ 
rent ” or the “equivalent single-phase current” of the three- 
phase circuit. 


14 


ALTERNATING CURRENT MOTORS. 


Equivalent Single-Phase Resistance. 

In determining the copper loss of a given piece of polyphase 
apparatus, it is convenient to know what value of resistance 
must be taken so that the square of the total current may, by 
its multiplication therewith, give the actual copper loss when 
the corresponding current flows in the circuit. The following 
very simple ratio is found to exist between the resistance of a 



Figs. 10, 11 and 12.—Determination of Equivalent Single¬ 
phase Resistance. 


given circuit as measured by direct-current instruments and 
the equivalent resistance for the total current: 

For any two-phase receiver with independent, star, three- 
wire, or mesh-connected coils, or for any three-phase receiver 
with delta, star or combination-connected coils, the equivalent 
resistance for the total current is equal to one-half of the value 
measured between phase lines by direct-current instruments. 



Figs. 13, 14 and 15.—Determination of Equivalent Single¬ 
phase Resistance. 


The proof of the above fact for circuits connected as shown 
by Figs. 10, 11 and 12, is self-evident and no explanation need 
be given. 

Referring to Fig. 13 let r = resistance per quarter; then 
2 r-f-2 = r = R resistance measured by direct-current instru¬ 
ments. Let i = current per coil; then y /2 Xi = current per 
lead and 2 y/% Xi = total current. Evidently the copper loss 




































SINGLE-PHASE AND POLYP HASE CIRCUITS. 


15 


is 4 Dr. Using the total current as above and R ~2 as the 

equivalent resistance the copper loss is (2 V 2 Xi) 2 Xr -r-2 = 4 i 2 r. 

In the delta-connected circuit of Fig. 14, let r = resistance per 
coil; then 2 r-r- 3 = 7? = resistance measured. Let i = current 
per coil; then 3 i = total current, and the copper loss is 3 i 2 r. 


For equivalent single-phase quantities, the loss is 


(3 i) 2 2 r 
2X3 


= 3 i 2 r. 

In Fig. 15 let r = resistance per coil; then 2 r = R = resist¬ 
ance measured, and if i = current per coil, then \/3" i = “ total ’’ 
current. The copper loss is 3 i 2 r. Using “ total ” current and 
equivalent resistance, the loss is (\/3~X7) 2 r-= Zi 2 r. 

Since the ratio of effective resistance for the equivalent 
single-phase current to the measured resistance is the same for 
star and delta-connected three-phase circuits, the same ratio 
must evidently hold for a combination of the two connections. 


Advantages of Employing Equivalent Single-Phase Quan¬ 
tities. 

The advantages derived from the consistent use of “ equivalent 
single-phase ” quantities reside not only in the simplicity in 
calculation, but also in the elimination of all ambiguity of 
expression. Thus when equivalent single-phase quantities are 
not used, the statement, “ the current at full load is 10 am¬ 
peres,” may mean 10 total amperes, 10 amperes per lead wire, 
or 10 amperes per phase wire. In other words, the real equiva¬ 
lent single-phase current might be 5.8 amperes, 10 amperes, 
17.3 amperes, 20 amperes, or 30 amperes. It is evidently far 
preferable to state, for instance, that the “ equivalent single¬ 
phase current ” at full load is 17.3 amperes. Likewise the 
statement, “ the primary resistance is 10 ohms,” could mean 
the resistance between phase lines, the resistance between ad¬ 
jacent lines in a mesh-connected two-phase winding, the re¬ 
sistance per delta-connected phase winding, or the resistance 
from line to neutral point in a star-connected winding. “ An 
equivalent single-phase resistance of 10 ohms,” however, can 
have only one interpretation. 



CHAPTER II. 

OUTLINE OF INDUCTION MOTOR PHENOMENA. 

Methods of Treatment. 

For dealing with the phenomena of induction motors, there are 
numerous points from which the problem may be viewed, each 
view point involving a certain method of treatment, but it may 
be stated that in general all methods lead to practically the 
same results. Thus the machine may be treated as a trans¬ 
former, or it may be considered a special form of alternating- 
current generator delivering current to a fictitious resistance 
as a load. It may be assumed that its torque is due to the 
current produced in the secondary of the transformer of one 
phase acting upon the magnetism due to the primary of another 
phase, or it is possible to consider that the magnetisms due to 
the separate phases combine to produce a revolving field in 
which the secondary circuits are placed. In what follows, the 
induction motor will be looked at from several points so that the 
reader will be able to obtain a clear view of the actions of the 
machine, a systematic attempt being made to present the motor 
in such a light as to avoid all unnecessary complexities which 
might tend to blur the vision. 

Before dealing with the complex inter-relation of the com¬ 
ponent parts of the motor which so combine in their actions as 
to produce the performance obtained from the structure, it is 
well first to take a glance at the machine in its simplest form 
so as to see just what may be expected from it. It is believed 
that this method, without introducing inaccuracies incon¬ 
sistent with the object sought, possesses the advantage of allow¬ 
ing the reader to become quickly acquainted with the machine, 
and permits him to ascertain the involved electromagnetic 
phenomena and to study them singly, and then conjointly, in 
the simplest possible manner. 

The machine that will be discussed here is the ordinary poly¬ 
phase induction motor, having a stationary primary wound 

16 


OUTLINE OF INDUCTION MOTOR PHENOMENA. 


17 


with overlapping coils in slots, and a revolving secondary which 
moves in the rotating field set up by the primary windings. 
It may be well at this point to call attention to the fact that 
the primary coils can occupy either the moving or the stationary 
member, provided that the stationary or the moving member, 
respectively, is wound with the secondary coils. Due to this 
fact the terms “ armature ” and “ field ” members, when ap¬ 
plied to an induction motor, are apt to lead to confusion. It 
should be noted that the machine has two windings, the “ pri¬ 
mary ” and t*he “ secondary,” either of which may be placed 
on the “ rotor ” or the “ stator.” 

Production of Revolving Field. 

As usually built, the primary coils form a distributed winding 
similar to that of a direct current armature. These coils are 
connected into groups according to the number of phases 
and poles for which the machine is designed, there being one 
group per phase per pole. 

The current for each phase is run through the corresponding 
group of each pole, tending to make alternate north and south 
magnetic poles around the machine. These north and south 
poles of each phase combine with those of the other phases to 
make resultant north and south poles. These resultant poles 
shift positions with the alternations of the currents and occupy 
places on the primary core determined by the relative strengths 
of the currents in the different groups of coils. Each magnetic 
pole, therefore, advances by the arc occupied by one group of 
windings for each phase during each alternation of the current. 
It is thus seen that the resultant magnetic field revolves at a 
speed depending directly upon the alternations and inversely 
upon the number of poles. 

In the revolving field is situated the secondary, in the windings 
of which is generated an e.m.f. determined by the rate at which 
its conductors are cut by the magnetic lines from the primal^. 
If the circuits of the secondary be closed there will flow therein 
a current which, being in a magnetic field, will exert a torque 
tending to cause the secondary member to turn in the direction 
of the revolving field. The final result is that the speed of 
the rotor increases to a value such that the relative motion of 
the secondary conductors and the revolving field generates 


18 


ALTERNATING CURRENT MOTORS. 


an e.m.f. sufficient to cause to flow through the impedance of 
the conductors a current the product of which into the strength 
of the field will equal the torque demanded. 

As can be seen, the speed of the rotor can never equal the 
speed of the rotating magnetic field, because the conductors of 
the secondary must cut the lines of force of the field in order 
to produce current in the secondary winding, and pull the rotor 
around; if the speeds were identical there would be no cutting. 
The difference between these two speeds is commonly known 
as the “ slip.” For the purpose of this discussion the slip will 
be considered in terms of the synchronous speed, that is to 
say, if the synchronous speed were 1200 r.p.m. and the actual 
speed of the rotor or secondary member were 1176 r.p.m. and 
the slip would be 24 -h 1200 = 0.02 

Since the conductors of the secondary cut an alternating field, 
first of one polarity and then of the other, the current produced 
in them will be alternating. The frequency of this current is 
normally much lower than the frequency of the supply current; 
the secondary frequency is equal to s f, s being the slip and 
/ the frequency. The current in the secondary being alter¬ 
nating, the impedance of the secondary winding has a magnetic 
leakage reactive component in addition to its resistance. The 
leakage reactance at standstill is, of course, equal to 

X 2 — 2 7 z f L 2 , 

/ being the frequency of the supply current and L 2 the coefficient 
of local self-induction of the secondary winding. The secondary 
reactance when the machine is in operation is equal to 

s X 2 = 2 7i } L 2 s 

s being the slip, as previously explained. 

It is evident that the reactance of the secondary increases 
directly with the slip, so that with a constant value of secondary 
resistance the impedance increases as the slip increases; not in 
direct proportion, however. The effect of the reactive com¬ 
ponent of the secondary impedance is to cause the secondary 
current to lag behind the secondary e.m.f. by a “ time-angle,” 
the cosine of which is equal to the resistance of the circuit di¬ 
vided by its local impedance. If 0 represents this angle, cos 0 
represents the power factor of the secondary current. 

Outline of Induction Motor Phenomena. 

Looking further into the effect of increasing the load on the 


OUTLINE OF INDUCTION MOTOR PHENOMENA. 


19 


motor, it will be plain that when the rotor is running near 
synchronism the reactance is of negligible effect, so that the 
secondary current increases directly with the slip and has a 
pow T er factor of approximately unity. As the load increases, the 
torque must be greater; the motor slip therefore increases, w r ith 
a consequent increase of secondary e.m.f. As the reactance 
begins now to increase, the impedance also increases, and the 
secondary current is no longer proportional directly to the 
secondary e.m.f., and the current which does flow has a power 
factor less than unity, that is to say, it is out of time-phase with 
the secondary e.m.f. Thus it is out of time-phase with the re¬ 
volving magnetism, thereby requiring a proportionately larger 
current to produce a corresponding torque. 

The increased draft of current demanded for the primary 
circuit entails a drop of e.m.f. in the primary windings, and, 
since there must be mechanical clearance between the primary 
and secondary cores, the lines of force which pass from the 
primary to the secondary are diminished by the increase of the 
secondary current. Therefore, the magnetic field which causes 
the secondary e.m.f. decreases with increase of load. Since only 
the qualitative behavior of induction motors is necessary for 
the purpose of this discussion, and no regard need be had at 
present for the quantitative performance, this effect will be 
momentarily neglected. The factor here neglected are treated 
at great length in subsequent chapters. 

Simple Analytical Equations 

Let E 2 = secondary e.m.f. at standstill; 
s E 2 = secondary e.m.f. at a slip s] 

X 2 = secondary reactance at standstill; 

5 X 2 = secondary reactance at a slip s; 

R 2 = secondary resistance, 
then \/A, 2 + s 2 X 2 = secondary impedance, 

77 * R<y 

I = — 5 2 — = secondary current, and cos 6 = 2 == 

VR 2 2 + s 2 X 2 2 v^ + s 2 X 2 2 

= secondary power factor; all three at the slip, 5. 

n __ 5 ^2 _ v — 2 - v K = secondary torque, at 

VR 2 2 + s 2 X 2 2 VRf + s 2 X 2 

the slip, 5 ; where AT is a constant depending upon the terms in 











20 


ALTERNATING CURRENT MOTORS. 


which torque is expressed, and upon the magnetic field,—the 
latter being here assumed constant. 

R~ s E , K 


Hence D = 


rotor torque at the slip 5 . 


R 2 2 + s 2 X 2 

An examination of these formulas reveals many of the char¬ 
acteristics of the induction motor, which it is well to discuss 
at this point. 

( 1 ) The torque becomes maximum when R 2 = s X 2 . 

(2) If R 2 be made equal to X 2 , maximum torque will occur 
at standstill. 

(3) Since at maximum torque R 2 = s X 2 the torque is equal to 


s 2 X 2 E 2 K 
2 s 2 X 2 2 


E 2 K 
2 XX 


and the value of maximum torque is independent of the resist¬ 
ance. 

5 E. 


(4) When X 2 is negligible the torque = K 


R. 


or the 


torque is directly proportional to the slip near synchronism, 
and inversely to the resistance. 

ERE 

(5) At standstill the torque = 2 ■ 2 y'\ which, when R 2 is 

1\ 2 "T ^2 

less than X 2 can be increased by the insertion of resistance up 
to the point where the two are equal; further increase of resist¬ 
ance will then decrease the torque. 

( 6 ) The starting torque is proportional to the resistance and 
inversely proportional to the square of the impedance. 

(7) Since power is proportional to the product of speed and 
torque, the output is equal to 

K R s E 

P = A (1 — s) - -3 = A (1 — s) D, and is a maximum 

■K- 2 T $ A- 2 

at a slip less than that giving maximum torque. 

5 E , 


( 8 ) 


/ = 


VR* + s* A ’, 2 

and at maximum torque R 2 2 = s 2 X 2 2 . Therefore, I at maximum 


5 E 2 E 2 


torque = — ^ ^ \ whence it is plain that the secondary 

current giving maximum torque is independent of the secondary 
resistance- 












OUTLINE OF INDUCTION MOTOR PHENOMENA. 


21 


(9) At maximum torque the secondary power factor = 
= 0.707. 


1 

V2 


( 10 ) 


I 2 R 2 = 


5 2 E 2 2 R 2 
R 2 + s 2 X 2 


s E 2 D 
~K~ 


and since E 2 is constant 


for constant impressed pressure, the copper loss of the secondary 
is proportional to the product of the slip and torque. 

(11) For a given torque the slip is proportional to the copper 
loss of the secondary and independent of the secondary reactance 
or coefficient of self-induction. 

(12) Output is equal to P = I E 2 (1 — s) cos Q 


_ s E 2 

P = — - 2 - . 

VR 2 2 + s 2 X 2 


E 2 (1 - s). 


R. 


VR 2 + s 2 X 2 


R 2 s E 2 (1-Q 
R 2 + s 2 X 2 


D_ 

K 


E 2 (1-5) 


(13) If all losses except that of the secondary copper be 
neglected, the input will be 


P + P R 2 = Te 2 (1 -s)+ DE2S 


K 


and the efficiency will be 


§ E 2 (l-s) 


D D E^ s 

k E 2 (1-s) + -g 


— = 1—5. 


which means that the efficiency is equal to the absolute speed 
in per cent, of synchronism. Since there must be losses in 
addition to that of the secondary copper, the efficiency is always 
less than the speed in per cent, of synchronism. 

(14) At a given slip the torque varies as the square of the 
primary pressure. This is seen from the fact that the sec¬ 
ondary current at a given slip will vary directly as the strength, 
of field, the power factor will remain constant, and therefore the 
torque which is obtained from the product of secondaiv cunent, 
power factor and field will vary as the square of the field. Since 
the strength of the field varies directly with the primary pressure 
at a given slip, the torque will vary as the square of the primary 

pressure. 














22 


ALTERNATING CURRENT MOTORS 


Starting Devices for Induction Motors. 

Having thus determined the behavior of the induction motor 
under various changes of condition, the next step is to ascertain 
the methods by which it can be adapted to services of different 
requirements. 

If the impedance of the secondary be sufficiently great to 
prevent a destructive flow of current when full pressure is ap¬ 
plied to the primary with the rotor at standstill, the motor 
may be started by being connected directly to the supply cir¬ 
cuit. This is a method adopted quite extensively for motors 
of small size. If the impedance consists for the most part of 
reactance, the current drawn will have a low power factor, 
which, in general, affects materially the regulation of the sys¬ 
tem; the starting torque will be small. 

It was proved above that by so proportioning the secondary 
resistance that R 2 = X 2 the maximum torque would be exerted 
at standstill. With resistance of such a value permanently 
connected in the secondary circuit, at full load the slip would be 
enormous and the efficiency correspondingly low, as shown 
above. As a compromise in this respect, motors are usually 
constructed with only comparatively large resistance in the 
secondary windings. 

In order to reduce the current at starting, large motors are 
often arranged to be supplied with a lower primary e.m.f. and 
the full line e.m.f. applied only after they have attained a fair 
speed. This is also especially desirable when otherwise the 
excessive starting torque would produce mechanical injury to the 
shafting, etc., driven by the motor. For elevators and cranes 
this method has found extensive application. Crane motors 
are built with a secondary resistance which gives maximum 
torque at standstill. The reduced primary pressure is secured 
either from lowering transformers or from compensators (auto¬ 
transformers). In either case, loops are taken to a suitable 
controller, by which the pressure supplied to the primary of the 
motor is governed. Pressures of a number of different values 
are thus obtained. The very intermittent character of the 
work performed by crane motors renders their low efficiency a 
necessary evil, since in any case the efficiency must be less than 
the speed in per cent, of synchronism. 

In order to combine large starting torque with good running 


OUTLINE OF INDUCTION MOTOR PHENOMENA. 


23 


speed and efficiency, it is customary to provide resistance ex¬ 
ternal to the secondary windings and arrange to suitably reduce 
this resistance as the speed increases, allowing the motor to 
run without external resistance for the highest speed. For 
continuous service this method has proved highly satisfactory 
and gives the best running efficiency. Ordinarily, this necessi¬ 
tates the use of collector rings for the revolving member, whether 
such be the primary or the secondary. When the secondary 
revolves it is universally given a three-phase winding, inde¬ 
pendent of the number of phases of the primary, and therefore 
three collector rings are used. The external resistance is con¬ 
nected either “ delta ” or “ star,” and regulated by a suitable 
controller. 

A very ingenious device for automatically adjusting the ex¬ 
ternal resistance was exhibited at the Paris Exposition. It 
was shown above that the frequency of the secondary e.m.f. 
depends directly upon the slip and is a maximum at standstill. 
If an induction coil of low resistance and high inductance be 
subjected to an e.m.f. of varying frequency, the admittance 
will vary inversely with the frequency. In the Fischer-Hinnen 
device, just referred to, there is connected external to the sec¬ 
ondary windings a non-inductive resistance of a value to give 
practically maximum torque at standstill. In parallel with 
this is connected a highly inductive coil of extremely small 
resistance. At zero speed the admittance of the external im¬ 
pedance consists practically of only that of the non-inductive 
resistance. As the speed increases the admittance increases, 
due to the decreased reactance, and at synchronism the external 
impedance is somewhat less than the resistance of the inductive 
coil. .The action is, therefore, in effect the same as though the 
external resistance had been decreased directly as the speed 
increased. It is claimed that the starting device occupies so 
small a space that it can be placed in the rotating armature, 
so that no collector rings are required. 

Concatenation Control. 

A common defect in all methods of speed regulation thus far 
described is the low efficiency at speeds far below synchronism. 
Attention has been frequently directed to the fact that the effi¬ 
ciency is in any case less than the speed in per cent, of syn- 


24 


ALTERNATING CURRENT MOTORS. 


chronism. In order, therefore, to run at high* efficiency at 
reduced speed, it is necessary to adopt some method of de¬ 
creasing the synchronous speed. 

The synchronous speed of a motor is equal to the alternations 
of the system divided by the number of poles of the motor. 
If, therefore, a motor is arranged for two different numbers of 
poles it will have two different synchronous speeds. While 
offering many advantages over other more complicated systems 
of speed control, and introducing no unsurmountable difficulties 
in practical application to existing types of motors, this method 
has as yet passed little beyond the stage of experimental in¬ 
vestigation. 

With mechanical connection between two motors, “ con¬ 
catenation ” or “ tandem ” control also offers a means of reducing 
the synchronous speed. In practice, the secondary of one motor 
is connected to the primary of the other, the primary of the 
first being connected to the line. The frequency of the current 
in the secondary of any motor depends upon its slip, as noted 
above. At standstill, therefore, the frequency impressed upon 
the primary of the second motor will be that of the line. As 
the motor increases in speed the frequency of the secondary 
of the first motor decreases, and at half speed the frequency 
impressed upon the primary of the second motor will be equal 
to the speed of its secondary; that is, it will have reached its 
synchronous speed. If now, the primary of the second motor 
be connected to the supply circuit and the secondary of the first 
motor be connected to a resistance, the motors will tend to in¬ 
crease in speed up to the full synchronism of the supply. 

By the use of suitably selected resistances for the secondary 
circuits of the two motors, this method gives the same results 
as the series-parallel control of direct-current series motors, with 
one important distinction, however. The series-wound motors 
tend to increase indefinitely in speed as the torque is diminished, 
while the induction motors tend to reach a certain definite speed, 
above which they act as generators. In this respect they re¬ 
semble quite closely two shunt motors with constant field ex¬ 
citation, having armatures connected successively in series and 
in parallel, with and without resistance, and, like the shunt 
motors, when driven above the normal speed they feed power 
back to the supply. 


CHAPTER III. 


OBSERVED PERFORMANCE OF INDUCTION MOTOR. 

Test with One Voltmeter and One Wattmeter. 

In the preceding chapter, a hasty glance was taken at the 
characteristics of an induction motor under certain assumed 
ideal conditions. It is well to show from actual tests just how 
closely the observed results compare with the ideal calculated 
results. Below there is given, therefore, the complete test of 



Fig. is. —Test of Three-phase Motor with One Voltmeter 

and one Wattmeter. 


an induction motor under actual operating conditions, and in¬ 
cidentally mention is made of a convenient method for testing 
a motor when the supply of instruments is limited. 

The method of testing a transformer or a generator by sep¬ 
aration of the losses is well known. It is such a method, which 
by a few slight modifications can be applied to induction motors, 
that is given below. Most of what is stated in this connection 
is true for any induction motor under any condition of service, 

25 



























26 


ALTERNATING CURRENT MOTORS. 


though the greatest simplicity in testing and the requisite use 
of the least number of instruments will be obtained only with 
polyphase motors operating on well balanced and regulated 
circuits. Each element of a test is treated separately. 

Measurement of Slip. 

The determination of the slip of induction motor rotors 
by counting the r.p.m. of both generator and motor is 
open to many objections. If the two readings of speed be not 
taken simultaneously though the true value of each be correctly 
observed, when the speed of either is fluctuating the value of 
slips will be greatly in error. A slight proportional error in 
either speed introduces an enormous error in the slip. Where 
the generator is not at. hand the above method obviously cannot 
be directly applied, and is applicable only when a synchronous 
motor is available for operation from the same supplv system 
as the induction motor. 

When the secondary current can be measured and the secondary 
resistance is known, the most accurate and convenient method 
for determining the slip is from the ratio of copper loss of sec¬ 
ondary to total secondary input. 

If we let I 2 be any observed value of secondarv current A, 

1 2 

the secondary resistance and W 0 the output of the motor, then 

gj.p _ s _ I 2 ^2 _ Copper loss of secondary 

II o -f 1 2 R 2 total secondary input 

As a proof of this fact, consider the magnetism cut by the 
secondary windings to be of a strength which would cause to 
be generated E 2 volts in the windings at 100 per cent. slip. 

Let 5 be any given slip, W s the total secondary watts, 0 the 
angle of lag of secondary current, and X 2 the secondary re¬ 
actance at 100 per cent, slip, then 

h 2 R 2 / 2 2 ^ 2 = / 2 2 R 2 

W 0 + P R 2 W s I 2 E 2 cos O' 



s E 2 

Vr*+x*s* 


s E 2 

R 2 sec 6 ' 


and 


U 2 r 2 

I 2 E 2 cos 0 


s I 2 E 2 R 2 

I 2 E 2 R 2 cos 6 sec 0 













OBSERVED PERFORMANCE OF INDUCTION MOTOR. 27 


Since neither E 2 nor X 2 appears in the above equation, the 
relation is independent of the strength of field magnetism cut 
by the secondary windings and of the secondary reactance. 


Determination of Torque. 


The torque of the rotor can be ascertained with the 
highest degree of accuracy and facility from the ratio of sec¬ 
ondary input to the synchronous speed, that is, the torque is 
expressed in pounds at one-foot radius by the following equa¬ 
tion: 


Torque = D = 7.04 


W P - L P 
syn. speed 


where W P is the total primary input; L P the total primary 
losses, and the synchronous speed is in r.p.m. 

If L P includes the friction, D will equal the external rotor 
torque, while if L P includes only the true primary iron and 
copper losses, D will be the total rotor torque. 

To prove the above expressed relation, let W s be total sec¬ 
ondary input, W Q the motor output, and 5 the rotor slip. Then 

33 000 W 0 
2 7 r r.p.m. 746’ 


but W 0 = 
Therefore, 


(1 — s) and r. p. m. = synchronous speed (1 — s ). 


D = 7.04 


W s 

syn. speed’ 


Therefore, for a given primary input and primary losses the 
rotor torque is independent of secondary speed or output, and 
any error in the determination of either of the latter quantities 
need not affect in the least the value obtained for the torque. 

If the total power received by the secondary is used up in the 
secondary resistance the equation for the torque will be 


7 2 R 

D 0 = 7.04- 1 -■ r 

syn. speed 


or starting torque, which, as has been stated previc :sly, may be 
increased by any method which will increase the stationary 
secondary copper losses. 







28 


ALTERNATING CURRENT MOTORS. 


Measurement of Secondary Resistance. 

Due to the inability to insert measuring instruments in the 
secondary of squirrel-cage induction motors, the ordinary 
direct-current method of determining the resistance cannot be 
used. 

If the rotor be clamped to prevent motion a wattmeter 
placed in the primary circuit will read the copper loss of both 
primary and secondary when the e.m.f. across the leads is re¬ 
duced to give a fair operating value of primary current. If 
from the reading of watts thus obtained there be subtracted 
the known copper loss of the primary for the current flowing, 
the value of the secondary copper loss will be secured. The 
resistance of the secondary (reduced to primary) will be ob¬ 
tained by dividing this loss by the square of the primary current. 
It is to be noted that certain minor effects are here neglected. 
These effects are of small moment and do not seriously modify 
the final results. However, they will be treated at length in a 
later chapter. 

Determination of Secondary Current. 

When the motor is provided with a squirrel cage secondary 
winding, measurement of the secondary current cannot be 
made directly, but the determination of its value must be 
by calculation. The variation in value and phase of the 
primary current may serve as an indication of the current 
flowing in the secondary windings, though the actual increase in 
primary current does not represent the increase in secondary 
current. 

The difference between the power components of the primary 
current at no load and under a chosen load may be taken as the 
equivalent increase in the power component of the secondary 
current under the same conditions, while the difference between 
the quadrature components of the primary current at the same 
time is a measure of the equivalent increase in the quadrature 
component of the secondary current. 

When the no-load value of secondary current is negligible 
the vector sum of the above-found components represents 
the secondary current for the chosen load in terms of the 
primary current. These facts will be discussed more fully 
hereafter. 


OBSERVED PERFORMANCE OF INDUCTION MOTOR. 29 

Calculation of Primary Power Factor. 

For a single-phase motor the determination of the primary 
power factor will usually involve the measurement of primary 
watts, volts and amperes. 



For two-phase motors with equal e.m.fs. across the separate 
phases, one wattmeter alone may be used to obtain the power 
factor by simply transferring the pressure coil from one phase 




















































































































30 


ALTERNATING CURRENT MOTORS. 


to the other, leaving the current coil always in one lead. The 
reading of the wattmeter in one case will be IV x = I h E cos 0 , 
wffiere I L is the current in each line wire; and in the second case, 


W 2 = I l E sin 0\ whence tan 0 = 


W 2 


from which 


may be ob¬ 


tained the power factor. 

For three-phase motors one wattmeter can similarly serve to 
indicate the power factor. The wattmeter readings will be 
W l = EE cos {0 — 30); W 2 = I h E (cos (9 + 30), 


whence tan 0 = 



W, -W 2 
W t + W 2 


Calculation of Primary Current. 

If the primary electromotive force, watts and pow T er factor 
be known, the primary current can readily be calculated and, 
therefore, need not be measured. 

It is evident from the above discussed facts that with two 
and three-phase induction motors operated from circuits having 
constant and equal e.m.fs. across the separate phases, one watt¬ 
meter and one voltmeter can be used to determine, primary 
watts, amperes and volts, and secondary amperes, and that 
when the primary resistance is known or can be measured the 
complete performance efficiency, etc., of the motors can at once 
be calculated. 

In the accompanying table and in Fig. 17 there are given the 
calculations and the curves of such a test made upon a 5-h.p., 
eight-pole, 60-cycle, three-phase induction motor. The equiv¬ 
alent single-phase primary resistance at running temperature 
was .078 ohm, and the equivalent secondary resistance (found 
as above) was .28 ohm. 

Since the copper loss of a three-phase receiver is expressed by 
the quantity I 2 R where I and R are equivalent single-phase 
values, for either star or delta-connected receiver, no attention 
need be paid to the method by wffiich the primary coils are 
interconnected within the motor or, in fact, wffiether the sec¬ 
ondary be wound delta, star or squirrel cage. In both the table 
and in Fig. 17, the current referred to is the “ total ” current 
or the equivalent single-phase current of the machine. 

It will be observed from column (14) of the table that 275 




THREE-PHASE INDUCTION MOTOR, PERFORMANCE TEST. INSTRUMENTS: ONE WATTMETER, ONE VOLTMETER. 



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32 


ALTERNATING CURRENT MOTORS. 


watts has been added to the primary copper loss to obtain the 
total primary loss. The value 275 represents the so-called 
“ fixed ” losses and is found by subtracting the known primary 
losses at “ no-load ” from the observed “ no-load ” input. 
The slight error occasioned by considering the “ fixed ” losses 
as constant will be discussed fully in a subsequent chapter. The 
constant 127.8 by which the total secondary input has been 
divided in column (16) to give the rotor torque is derived from 
consideration of the fact that an 8-pole motor operated at 60 
cycles has a synchronous speed of 900 r.p.m.; that is, 127.8 = 
900 -7- 7.04. From column (24) it will be noted that the speed has 
been taken as the ratio of the motor output to the total secondary 
input, that is, as equal to the secondary efficiency. This re¬ 
lation, which was discussed in a previous chapter, is exact, 
and it follows at once from the fact that the slip is equal to the 
ratio of the secondary copper loss to the total secondary input. 
These facts will be treated at length in a later chapter. 

Tests made upon this same motor by the output-input meth¬ 
ods agree throughout the whole range of the test with the 
herewith recorded test within the limits of the inevitable errors 
of observation of the various instruments used in the tests. 


CHAPTER IV. 


INDUCTION MOTORS AS FREQUENCY CONVERTERS. 

Field of Application. 

For the satisfactory operation of arc lamps a frequency higher 
than 40 p.p.s. is required, while when the frequency is much 
below this value incandescent lamps may show a fluctuation 
in brilliancy. The output of transformers may be shown to 
vary as the three-eighth power of the frequency. These are 
among the causes for the fact that in the older lighting stations 
a frequency as high as 133 p.p.s. was quite common. 

With later increase of magnitude and range of service, it was 
found that a lower frequency improved the operation of alter¬ 
nators in parallel, while the line regulation was also benehtted 
by the change from the higher frequency. This led to the adop¬ 
tion of a periodicity of about 50 to 60 p.p.s. With the advent 
of the long-distance power transmission circuits, the advisability 
of the adoption of a still lower frequency became apparent, 
while the successful use of rotary converters for railway work 
practically necessitates a frequency as low as 25 cycles. This 
is the present standard frequency for such service. 

When lighting is to be done from power supplied at 25 cycles, 
some device is usually provided for altering the nature of the 
current before it is applied to the lamps. A most satis¬ 
factory method of accomplishing this result is by means of alter¬ 
nating-current motors, of either the induction or synchronous 
type, driving lighting generators. Where only high-voltage 
alternating currents are available this method requires the use 
of step-down transformers, a motor and a generator, each 
carrying the full load. When the pressure at hand is suffi¬ 
ciently low, step-down transformers may be dispensed with, 
but a double equipment and double transformation of power 
is still necessary, with consequent low efficiency and high cost 

of installation. 

A convenient method for changing the lower frequency to a 

33 


24 


ALTERNATING CURRENT MOTORS. 


value suitable for lighting purposes is by the use of “frequency 
converters,” which constitute a special adaptation of induction 
motors as secondary circuit generators. 

Characteristic Performance. 

In the ordinary induction motor the frequency of the sec¬ 
ondary current is not that of the supply, but it has a value 
represented by the product of the slip of the rotor from syn¬ 
chronous speed and the frequency of the primary current. 
It is only when the slip is unity, or at standstill that the pri¬ 
mary and secondary frequencies are equal. Under this con¬ 
dition the windings are in a true static transformer relation, 
and, with the rotor clamped to prevent relative motion, current 
at the primary frequency can be drawn from the secondary 
windings. Obviously, the air-gap renders the induction motor 
for such purposes much inferior to a static transformer, on 
account of magnetic leakage between the coils. 

If, now, the secondary be given a motion relative to the 
primary, there may continue to be drawn from the secondary, 
current at a frequency determined by the slip from synchronous 
speed. If this slip be greater than unity—that is, if the motor 
be driven backwards—the frequency of the secondary current 
will be greater than that of the primary. By properly propor¬ 
tioning the rate of backward driving, the secondary current 
can be given a frequency of any desired value. 

In order that the secondary frequency may bear a constant 
ratio to the primary, it is necessary that the relative slip from 
synchronism be constant, which condition can conveniently 
be obtained by driving the rotor with a synchronous motor 
operated from the same supply system as the primary. 

Evidently the synchronous motor will demand power from 
the supply in addition to that demanded by the primary circuit 
of the frequency converter. The amount of this power is de¬ 
termined by the speed of the synchronous motor and the torque 
exerted by the frequency converter. When the slip of the 
converter is unity, the power demanded by the synchronous 
motor is zero; when, however, the slip is two—that is, when 
the frequency of the secondary is twice that of the primary— 
the power demanded by the synchronous motor is equal to 
that demanded by the primary of the converter. A further 


INDUCTION MOTORS AS FREQUENCY CONVERTERS, 35 

analysis will show that the power supplied by the frequency 
converter bears to the total output the ratio of the primary 
frequency to that of the secondary, the remaining power being 
supplied by the synchronous motor. 

Capacity of Frequency Converters. 

Since the total output appears at the secondary of the con¬ 
verter it behooves us to ascertain in what manner the power 
* supplied by the synchronous motor enters the converter wind¬ 
ings. 

Let the ratio of primary to secondary turns be unity, and let 
us consider the secondary current in phase with the secondary 
e.m.f., and let it be counter-balanced by an equal current in 
the primary in phase with its e.m.f., then 

I = primary current in phase with the primary e.m.f. 
and in phase opposition to the secondary current. 

I = secondary current. 

E = primary impressed e.m.f. 

S E = secondary generated e.m.f., where S = slip with 
synchronism as unity. 

IE = primary power x 

I S E = secondary electrical power. 

In the ordinary induction motor, where S is less than unity, 
the secondary generated power (SI E) is dissipated in the 
copper of the secondary windings, while the remaining power 
received from the primary (/ E - S I E) is available for mechan¬ 
ical work. When S E is equal to unity the secondary generated 
power (/ E) is totally available as electrical power, which may 
be lust in the resistance of the secondary windings or usefully 
applied to external work, while the mechanical power of the 
secondary is zero. When S is greater than unity the total 
secondary generated power (5 l E) is available at the secondary 
terminals. Of this power (/ E) is supplied by the primary, 
while the remainder is supplied by the synchronous motor. 

It is seen, therefore, that the effect of driving the secondary 
backwards is to increase the secondary pressure above that of 
the primary, and that the power for such increase is derived 
from the synchronous motor. 

A moment’s reflection will show that there is only partial 
double transformation of power with a frequency converter, 


36 


ALTERNATING CURRENT MOTORS. 


and that the sum of the capacities of the synchronous motor 
and the converter must just equal the output plus the inevitable 
losses in each machine. A numerical example will show this 
quite plainly. If we assume a lighting load of 60 kilowatts to 
be changed in frequency from 25 to 60 cycles, then the capacity 
of the frequency converter proper must be 25 kilowatts, and 
that of the synchronous motor 35 kilowatts, as shown above. 
It should be noted, however, that while the iron loss of the con¬ 
verter primary is the same as that of a 25-kw. induction motor 
on 25 cycles, the iron loss of the secondary of the converter is 
that of a 60-cycle, 60-kw. generator, and thus very materially 
greater than that of a 25-kw. induction motor, which latter, 
in fact, is usually quite negligible. 

By over-excitation, the leading component of the current de¬ 
manded by the synchronous motor may be adjusted to equal 
the lagging component due to the exciting current of the fre¬ 
quency converter, so that the external apparent power factor of 
the equipment may be kept quite high. 

Although very little practical use has as yet been made of 
the frequency converter in the simple form discussed above, 
the machine is employed extensively m combination with a 
synchronous (rotary) converter for delivering constant poten¬ 
tial direct current when the supply is high-voltage alternating 
current. The combination machine is termed a “ cascade ” 
converter or a “ motor converter.” 

Motor Converters. 

The motor converter consists of two machine structures whose 
revolving parts are mounted on the same shaft. The input 
machine resembles in every respect an induction motor with a 
coil-wound (rotor) secondary; the output machine is exactly 
similar to a rotary converter, and receives its current from the 
secondary winding of the input induction motor at a frequency 
much reduced from that impressed upon the primary winding. 
A diagram of the operating circuits of the motor-converter is 
shown in Fig. 18, the stationary field windings of the output 
machine being omitted in order to avoid unnecessary compli¬ 
cations. 

If it be assumed that the input machine has the same number 
of poles as the output machine, then the operating speed of the 


INDUCTION MOTORS .45 FREQUENCY CONVERTERS. 37 


set is equal to just one-half of the speed of the revolving field 
of the first machine (induction motor). Consider the action 
of the stator circuits of the input machine. When a certain 
alternating e.m.f. is impressed upon its terminals, the flux 
must have a value such that its rate of change produces a counter 
e.m.f. only slightly less than the impressed. When the primary 
(stator) circuits are symmetrically arranged and subjected to 
polyphase electromotive forces, the familiar synchronously re¬ 
volving field is produced. This field cuts across the secondary 
(rotor) conductors and generates therein electromotive forces 
having a frequency proportional to the slip from synchronous 



speed. The polyphase electromotive forces counter generated 
in the armature of the output machine due to its motion through 
the constant stationary field, will be proportional directly to 
the speed. It is evident that these two polyphase electromotive 
forces will have the same frequency at a certain speed of the 
revolving member, which speed will be one-half of that of the 
revolving field of the induction motor when the poles of the 
input and output machines are equal in number. 

Division of Power Between the Motor and the Converter. 

It is interesting to investigate the sources of the power re¬ 
ceived by the output machine. A little study will show that 
the rotor reaches a definite speed at one-half of the speed of 



































38 


ALTERNATING CURRENT MOTORS. 


the revolving field, and that no change in “ slip ” can accompany 
a change in load, so that the ordinary phenomena connected 
with the performance of the polyphase induction motor are 
completely lacking. When current is drawn from the direct- 
current side of the output machine, a certain component of the 
power demanded is supplied by the action of current which 
flows through the secondary windings of the input machine. 
This current tends to alter the value of the core flux of the in¬ 
duction (input) machine, and a counter balancing component 
of current flows in the primary coil and restores the core flux 
to approximately its initial vaiue. Thus a portion of the 
power is transmitted by transformer action. The current in 
the induction motor secondary produces a torque on the rotor 
due to its presence in the core flux, and hence a portion of the 
power is transmitted mechanically through the shaft by motor 
action. The ratio of motor action to transformer action is 
the same as the ratio of the number of poles of the input machine 
to that of the output machine. 

The output machine is in reality a combined synchronous 
converter and a direct-current generator. Its power as a syn- 

i 

chronous converter is supplied from the frequency converter 
portion of the input machine, while the power to operate the 
direct-current generator part is derived from the synchronous 
motor portion of the primary machine. 

As connected in the circuit the input machine performs the 
function of a true induction frequency converter, the frequency 
of the current in the secondary depending directly upon the 
slip of the rotor from synchronism. A distinction between the 
performance of this machine and that of a simple induction 
motor is found in the fact that the “ slip ” is constant and inde¬ 
pendent of the torque. The current from the brushes of the 
output machine determines the load, and the torque demanded 
thereby is supplied simultaneously by the action of the sec¬ 
ondary current on the revolving field of the input machine, 
and by the action of the armature current on the field of the 
output machine. 

Advantages of the Motor Converter. 

. The advantageous features of the motor-converter as com¬ 
pared with a synchronous converter reside in the fact that it 


INDUCTION MOTORS AS FREQUENCY CONVERTERS. 39 


may be fed with alternating current of any frequency and at 
any electromotive force and that it will deliver direct current 
from a commutator operating at a frequency best adapted for 
satisfactory performance. As will be pointed out in a subse¬ 
quent chapter, these advantages are most pronounced when the 
supply frequency is very high, while they disappear at low 
frequency. 

The motor-converter is started up from rest as an ordinary 
polyphase induction motor, by means of the starting resistance 
shown in Fig. 18; “ synchronous ” speed of the rotor is readily 
determined • by use of a voltmeter connected across between 
the slip rings. In respect to its starting characteristics the 
motor-converter is superior to either a synchronous converter 
or a synchronous motor-generator set; it is the equivalent of an 
induction motor-generator set. It is preferable to either type 
of motor-generator set in that it is less expensive to construct, 
and is more efficient in operation. 

Excitation of Motor-Converters. 

Although the input machine is in reality an induction motor 
it possesses many of the characteristics of a synchronous ma¬ 
chine. The constancy of its speed, as though it were a syn¬ 
chronous motor, has already been mentioned; it possesses an¬ 
other feature which renders it similar to a synchronous machine, 
namely, the action of the wattless component of the load cur¬ 
rent upon its field magnetism is quite the same as though the 
machine were a synchronous generator. The e.m.f. in the 
secondary of the induction motor wffien the rotor is stationary 
bears to the e.m.f. of the primary the ratio of the respective 
turns of the secondary and the primary coils; this e.m.f. varies 
directly with the slip, and at each value of the slip it has a de¬ 
finite determinable value. If the input and output machines 
have equal number of poles, then the normal operating value 
of the secondary e.m.f. of the input machine is just one half of 
the value when the rotor is stationary. Since the internal op¬ 
erating e.m.f. of the output machine depends upon the strength 
of the stationary field, there is a definite field strength to corre¬ 
spond to the normal operating conditions of the equipment. 
Since the alternating-current side of the output machine is 
similar in all respects to a synchronous motor, any excess of 


40 


ALTERNATING CURRENT MOTORS . 


excitation of its field will cause the machine to draw from the 
source of supply a component of current leading with respect 
to the e.m.f., and of a value such as to restore the magnetism 
threading the armature approximately to its normal value. 
Such leading current flowing in the secondary of the input 
machine has the effect of assisting in supplying the exciting 
magnetomotive force of this machine and thus of decreasing 
the lagging exciting component of, the current in the primary 
windings, and thereby improving the power factor. 

i 

Phases of the Motor-Converter. 

On account of the fact that the secondary of the input ma¬ 
chine is directly tapped to the armature of the output machine, 
a considerable number of interconnections is only slightly more 
expensive than a lesser number. It is advantageous to employ 
a large rather than a small number of phases with a rotary 
converter or synchronous motor. Hence the revolving mem¬ 
ber of the motor-converter is frequently arranged for a great 
number of phases, such as 9 or 12, although the primary of the 
induction motor may be wound for only one, two or three phases. 

In a subsequent chapter the inherent characteristics of the 
rotary converter will be discussed at length. It is well at 
this point, however, to mention the fact that a polyphase 
rotary converter is preferable in every respect to a single-phase 
rotary converter. A single-phase motor-converter, on the other 
hand, compares very favorably with a polyphase motor-converter; 
it differs therefrom in respect merely to the primary winding of 
the input machine. It possesses excellent synchronizing force 
and has an operating efficiency that compares favorably with 
a similar polyphase motor-converter. 

Uses of the Motor-Converter. 

The advocates of the motor-converter claim that in comparison 
with a motor-generator the motor-converter is more economical 
in first cost and 2J per cent, more efficient in operation. In 
comparison with a rotary converter and the necessary bank of 
transformers, the motor-converter is about equally as expensive 
in first cost and has an efficiency 1 per cent, less than the rotary 
equipment. The motor-converter is claimed to be better than 
the rotary converter for frequencies above 40 cycles on account 
of the improved commutation at the low frequency used in the 


INDUCTION MOTORS *45 FREQUENCY CONVERTERS. 41 

direct-current portion of the machine. For lower frequencies, 
such as from 20 to 30 cycles, the rotary converter is evidently 
preferable. For all frequencies, however, the motor-converter 
possesses several characteristics which render it desirable even 
in comparison with rotary converters. Thus the motor-con¬ 
verter affords much better control of the voltage of the current 
delivered and it requires less skilled attention. 

The motor-converter has been introduced on a large scale 
for electric railway work in Great Britain, but has been prac¬ 
tically ignored in this country. The reason for the attitude 
of the American manufacturers and operating engineers with 
reference to the motor-converters is to be found in the fact that 
the conditions under which the machine operates most ad¬ 
vantageously as compared with the rotary converter seldom 
exist in this country. It is admitted by the advocates of the 
motor-converter that for use on 25-cycle polyphase circuits 
the machine is inferior to the rotary with reference to first cost 
and running expenses, although it possesses some desirable 
features in connection with its characteristics under starting 
and operating conditions. On account of the fact that a very 
large proportion of the power used with alternating-current 
converters in this country is obtained from 25-cycle circuits, it is 
apparent that the motor-converter must necessarily find a 
more limited field for application in America than in England, 
where a frequency of 50 cycles is often used for general power 

purposes. 


Improvement of the Power Factor of an Induction Motor 
by Utilization of its Frequency-Converter Prop¬ 
erties. 

The most logical method of improving the power factor of 
operation of an induction motor is that which has for its object 
the reduction of the wattless component of the piimar^ current. 
Although it is for the purpose of producing counter e.m.f. in 
the primary that the core magnetism is required, it is not es¬ 
sential that the exciting current should flow in the primary 
windings; the magnetomotive force for excitation may with 
equal effect be supplied by current in the secondary, although 
the ampere-turns in the one case must equal those in the other. 


42 


ALTERNATING CURRENT MOTORS. 


Direct Current in Secondary Coils. 

It is possible to 'supply the exciting magnetomotive force to 
the motor through the secondary windings while the rotor 
travels at full speed without any constructive change in the 
windings of the motor. Consider normal e.m.f. impressed 
upon the primary with the secondary on closed circuit and 
the rotor traveling at full speed. The wattless component of 
the primary current will have practically the same value as 
with the secondary on open circuit and the rotor stationary. 
If now there be introduced into the secondary windings direct 
current adjusted in value to equal the mean effective value of 
the wattless component of the primary current, as previously 
observed, it will be found that the lagging component of the 
current in the primary windings has been reduced to zero, so 
that the power factor has increased to unity. The motor may 
now be given its full load and its performance will be found 
to be satisfactory in all respects. 

With direct current supplied to its secondary windings an 
induction motor travels at synchronous speed, and, in fact, 
becomes transformed to a synchronous motor and therefore pos¬ 
sesses all of the qualities inherent in the performance of this 
type of machine. Mr. A. Heyland has devised a method, how¬ 
ever, by wdiich approximately unidirectional current may be 
supplied to the secondary windings while the motor yet retains 
the characteristics of the asynchronous type of machine. 

Alternating-Current in Secondary Coils. 

The method of obtaining this most desirable result consists 
in applying to the secondary windings a commutator to which 
current from the source of supply for the primary is led by 
w r ay of properly disposed brushes. Adjacent segments of the 
commutator are connected together by non-inductive resistance 
external to the windings, there is no possibility of sparking at 
the brushes, and the performance of the commutator is quite 
similar to that of slip rings. 

Fig. 19 represents diagrammatically a direct-current armature 
complete with commutator, to be used as the secondary of an 
induction motor. Since no current whatever will flow in the 
conductors of a direct current armature when on open circuit, 


INDUCTION MOTORS .45 FREQUENCY CONVERTERS. 43 

the armature alone will possess no tendency to be drawn into 
rotation by the revolving magnetism of the primary. In 
order that the armature winding may serve as the secondary 
of the induction motor, it is necessary that points on the ar¬ 
mature possessing difference of potential be joined together. 
The resistance, shown in Fig. 19 as being connected between 
adjacent segments, serves to complete the secondary circuit, 
and with the brushes removed from the commutator the motor 
thus equipped will operate in all respects similarly to one with 
a pure “ squirrel-cage ” secondary winding. The three brushes 
shown in the figure are for the purpose of allowing the intro- 



Fig. 19—Direct-current Armature of Heyland Motor for 
Three-phase Secondary Excitation, Showing Segment¬ 
connecting Resistances. 

duction of three-phase current into the secondary for supplying 
sufficient magnetomotive force for field excitation, in order 
that no wattless current need flow in the primary windings. 

According to the foregoing discussions it is plain that with 
the rotor traveling at synchronous speed, current of zero fre¬ 
quency may be utilized to supply the magnetomotive force for 
excitation, while with the rotor stationary, current at a fre¬ 
quency equal to that of the primary may thus be employed. A 
little further consideration will convince one that at speeds 
between synchronism and standstill current at intermediate 
frequencies may be so used, and that the requisite frequency 
in each case is equal to the product of the percentage of slip 









44 


ALTERNATING CURRENT MOTORS. 


and the primary frequency. By attaching to the secondary 
windings a commutator, to the brushes upon which current is 
led at the primary frequency, the current in the secondary will, 
at any speed, possess the frequency required for excitation. 

Action with Stationary Rotor. 

For the sake of simplicity in explanation, consider, in the first 
place, that upon the rotor there is placed a symmetrical direct- 
current armature winding with a corresponding commutator. 
By introducing into the windings an alternating current at the 


TABLE 

I.—Instantaneous values of currents in 
three-phase excitation; rotor 

each coil of 
stationary. 

direct-current 
(Fig. 19). 

winding; 

Number of 
coil. 

A 

B 

C 

D 

E 

F 

1 

+ 10.00 

+ 8.66 

+ 5.00 

0.00 

— 5.00 

—8.66 

2 

+ 10.00 

+ 8.66 

+ 5.00 

0.00 

— 5.00 

—8.66 

3 

+ 10.00 

+ 8.66 

+ 5.00 

0.00 

— 5.00 

—8.66 

4 

+ 10.00 

+ 8.66 

+ 5.00 

0.00 

— 5.00 

—8.66 

5 

— 5.00 

—8.66 

—10.00 

—8.66 

— 5.00 

0.00 

6 

— 5.00 

—8.66 

—10.00 

—8.66 

— 5.00 

0.00 

7 

— 5.00 

—8.66 

—10.00 

—8.66 

— 5.00 

0.00 

8 

— 5.00 

—8.66 

—10.00 

—8.66 

— 5.00 

0.00 

9 

— 5.00 

0.00 

+ 5.00 

+ 8.66 

+ 10.00 

+ 8.66 

10 

— 5.00 

0.00 

+ 5.00 

+ 8.66 

+ 10.00 

+ 8.66 

11 

— 5.00 

0.00 

+ 5.00 

+ 8.66 

+ 10.00 

+ 8.66 

12 

— 5.00 

0.00 

+ 5.00 

+ 8.66 

+ 10.00 

+ 8.66 


primary frequency, when the rotor is stationary, the effect will 
be exactly the same as was previously found when the current 
at the same frequency was supplied to the secondary of the 
ordinary induction motor with stationary rotor; that is, by 
adjustment of secondary current, the wattless component of 
the primary current may be made to disappear. If, however, 
the rotor be given a certain speed, the same current in the sec¬ 
ondary will continue to produce the same magnetizing effect as 
at standstill, for at each instant the commutator will cause 
the current to traverse conductors occupying the same position 







INDUCTION MOTORS AS FREQUENCY CONVERTERS. 45 

in space, and the effect of the secondary current upon the pri¬ 
mary core magnetism will not be altered. 



Fig. 20.—Currents for Fig. 21.—Currents for 
Stationary Armature. Armature at Syn¬ 
chronous Speed. 


Fig. 20 represents such symmetrical armature winding upon 
the commutator of which are placed three brushes for the in¬ 
troduction of three-phase current for excitation. The in- 





4(3 


ALTERNATING CURRENT MOTORS. 


stantaneous values of the current in the individual coils as the 
current in the three-phase leads changes value are indicated 
for each 30 degrees increment of time in the several diagrams 
of Fig. 20, and collectively recorded in Table I. The rotor is 
here assumed to be stationary. It will be observed that through¬ 
out each cycle the current in each coil undergoes a double 
reversal and reaches full normal value in both positive and nega¬ 
tive directions. The reactive e.m.f. induced in the windings, which 
is proportional to the rate of change of the local flux surrounding 
the conductor due to the current flowing therein, is therefore 


TABLE II.—Instantaneous values of currents in each coil of direct-current winding;, 
three-phase excitation; synchronous speed. (Fig. 20). 


Number of 
coil . 

A 

B 

c 

D 

E 

F 

1 

+ 10.00 

4 

0.00 

+ 5.00 

+ 8.66 

+ 10.00 

0 00 

2 

+ 10.00 

+ 8.66 

+ 5.00 

+ 8.66 

+ 10.00 

+ 8.66 

3 

+ 10.00 

+ 8.66 

+ 5.00 

+ 8.66 

+ 10.00 

+ 8.66 

4 

+ 10.00 

+ 8.66 

+ 5.00 

0.00 

+ 10.00 

+ 8.66 

5 

— 5.00 

+ 8.66 

+ 5,00 

0.00 

— 5.00 

+ 8.66 

6 

— 5.00 

— 8.66 

+ 5.00 

0.00 

— 5.00 

— 8.66 

7 

— 5.00 

— 8.66 

— 10.00 

0.00 

— 5.00 

— 8.66 

8 

— 5.00 

— 8.66 

— 10.00 

— 8.66 

— 5.00 

— 8.66 

9 

— 5.00 

— 8.66 

— 10.00 

—8 66 

— 5.00 

— 8.66 

10 

— 5.00 

0.00 

— 10.00 

— 8.66 

— 5.00 

0.00 

11 

— 5.00 

0.00 

+ 5.00 

— 8.66 

— 5.00 

0.00 

12 

— 5.00 

0 00 

+ 5.00 

+ 8.66 

— 5.00 

0.00 


of a value corresponding to the primary frequency, and there 
is required for the excitation a wattless component of e.m.f. 
equal to that which would have been required had the exciting 
current been allowed to flow in the primary windings. 

Action with Rotor at Synchronous Speed. 

When the rotor is traveling at a speed approximately syn¬ 
chronous the exciting current required in the secondary windings 
is of the same value as before, but the requisite wattless com¬ 
ponent is much reduced because of the decrease in the reactive 







INDUCTION MOTORS .45 FREQUENCY CONVERTERS. 47 


e.m.f. of the secondary windings. With an infinite number of 
commutator segments and of phases for the exciting current 
in the secondary, the reactive e.m.f. would entirely disappear 
at synchronous speed, and it would increase directly with the 
rotor slip. Fig. 21 indicates the changes in the value of the 
secondary current in the individual coils with three-phase 
excitation, when the rotor is traveling at synchronous speed, 
For the sake of clearness the windings are considered stationary 
and the brushes are supposed to revolve at synchronous speed, 
the effect being the same as with stationary brushes and re¬ 
volving windings, of course. 

A glance at Table II will show that the fluctuations in the cur¬ 
rent in the individual coils are much reduced at synchronous 
speed, and consequently the flux around the conductors, to the 
rate of change of which is due the reactive e.m.f., has a much 
more nearly constant value than when the secondary is sta¬ 
tionary, and thus the necessary wattless component for second¬ 
ary excitation is correspondingly diminished. 

The value of the e.m.f. for excitation will depend upon the 
number of and resistance of the secondary conductors and upon 
the reactive e.m.f., and will in general be much below 
that required for the primary windings. It can con¬ 
veniently be obtained by transformation from the supply 
circuit. An increase in the excitation e.m.f. above normal 
value will cause the primary to draw leading currents, the value 
of which may be adjusted to equal the lagging current de¬ 
manded by the primary of the excitation transformers, so that 
the motor and transformers considered as a unit may be oper¬ 
ated at unity power factor. 

It is scarcely probable that the mere elimination of the watt¬ 
less current component from the circuit wires would prove 
sufficient inducement to a consumer to justify the extra ex¬ 
pense of adding a commutator and the accompanying complica¬ 
tions to the one piece of reliable machinery the simplicity of 
which has previously been the characteristic that led most rap¬ 
idly to its adoption in preference to commutator motors. The 
ability of manufacturers to produce at a reduced cost motors 
giving satisfactory service to the purchaser can alone cause a 
compensated motor to compete successfully with its highly 
efficient and, above all, simple rival. 


CHAPTER V. 


THE SINGLE-PHASE INDUCTION MOTOR. 

Outline of Characteristic Features. 

While under no condition is the single-phase motor more 
satisfactory or economical than the polyphase machine, yet, 
by a little care in the selection of a motor for the service re¬ 
quired, the performance of the single-phase machine may 
compare quite favorably with that of the polyphase type. 
The most prominent difference between the single-phase and 
the polyphase motor is the inability of the former to exert a 
torque at standstill. Numerous devices have been applied to 
render single-phase motors self-starting, which have met with 
varying success. A difficulty which the designer has had to 
encounter lies in the fact that, with few exceptions, such de¬ 
vices are applicable only to motors of small sizes, or where 
efficiency is of small moment. Little trouble is experienced 
in designing self-starting single-phase motors for meter or fan 
work, but the problem assumes a different aspect when motors 
for power purposes are desired. 

In what follows, an attempt will be made to outline the 
characteristic features of the single-phase induction motor, to 
ascertain the similarities and the differences between the per¬ 
formance of a single-phase and that of a polyphase machine, 
and to investigate the methods by which the single-phase motor 
may be operated under various conditions. The graphical 
representation of the phenomena of the single-phase motor is 
reserved for a subsequent chapter. 

Although supplied with current, which, if acting alone, 
could produce only a simple alternating magnetism, in contra¬ 
distinction to a rotating field, it is found that a single-phase 
motor under operating conditions develops a rotating field 
essentially the same as would be obtained were the machine 
operated on a polyphase circuit. The effect of the mechanical 

48 


THE SINGLE-PHASE INDUCTION MOTOR. 


49 


motion of the secondary in producing the rotating field may 
be determined as follows: 

Production of Quadrature Magnetism. 

Assume for the purpose of illustration a motor with four 
mechanical poles having the two opposite poles excited by a 
single-phase alternating current, and consider the moment 
when one of these poles is at a maximum north and the other 
a maximum south, as shown in Fig. 22. If the rotor be moving 
across this field in the direction indicated, there will be gen¬ 
erated in each of the conductors under the poles an e.m.f. pro¬ 
portional to the product of the field magnetism and speed of 



Fig. 22. —Production of Quadrature Magnetism. 

rotor. Evidently if the speed be constant, of whatsoever 
value, this e.m.f. will vary directly with the strength of mag¬ 
netism; that is, it will be maximum when the magnetism is 
maximum, and zero at zero magnetism. Other conditions re¬ 
maining the same, the maximum value of the secondary e.m.f. 
will vary directly with the speed of the rotor. 

If the circuits of the rotor conductors be closed, there will 
tend to flow therein currents of strengths depending directly 
upon the e.m.fs. generated in the conductors at that instant 
and inversely upon the impedance of the rotor conductors. The 
somewhat unique condition of e.m.fs. in a combined series and 
parallel circuit, which.exists in the rotor as here described, is 
depicted by analogy in Fig. 23, where the e.m.f. in each conductor 
























50 


ALTERNATING CURRENT MOTORS. 


across the rotor core is represented by a battery. The abso¬ 
lute value of each e.m.f. depends upon the strength of the pri¬ 
mary field and the position of the conductor in that field, and 
hence changes from instant to instant. The current which 
flows through the end rings changes its direction of flow with 
reference to the field poles once for each reversal of the primary 
magnetism. 

The current which flows through the rotor circuits at once 
produces a magnetic flux which, by its rate of change in value 
generates in the rotor conductors a counter e.m.f., opposing the 
e.m.f. that causes the current to flow, and of such a value that 
the difference between it and this e.m.f. is just sufficient to 
cause to flow through the impedance of the conductors a current 
w r hose magnetomotive force equals that necessary to drive the 



Fig . 23.—Current and e.m.f. in Squirrel-cage Secondary. 


required lines of magnetism through the reluctance of their 
paths. Since this latter magnetism must have a rate of change 
equal (approximately) to the e.m.f. generated in the rotor 
conductors by their motion across the primary field, and since 
this e.m.f. is in time-phase with the primary field, it follows 
that this magnetism must have a value proportional to the rate 
of change of the primary magnetism, and, if the primary mag* 
netism follows a sine curve of values, this magnetism must 
follow the corresponding cosine curve; that is, it must be in 
quadrature to the primary magnetism as to time phase. 

Consider the N pole due to primary magnetism at its 
maximum strength, and decreasing in value. The e.m.f. gen¬ 
erated in the rotor will tend to send lines of force at right angles 
to the primary field, inducing a secondary N pole at the right 
(see Fig. 22). The lines of secondary magnetism (induced field) 

























THE SINGLE-PHASE INDUCTION MOTOR. 


51 


continue to increase in number so long as the primary mag¬ 
netism does not change direction of flow. They, therefore, 
reach their maximum yalue when the primary magnetism dies 
down to zero, at which instant the induced or secondary mag¬ 
netism will have its maximum strength with the north pole 
to the right, as drawn. 

As the primary magnetism now shifts its north pole to the 
bottom, the secondary lines begin to decrease in number, and 
they will reach their zero value when the primary magnetism 
reaches its maximum strength. The induced magnetism will 
then begin to increase its lines in the reverse direction, pro- 
ducing a north pole to the left, and it will reach its maximum 
strength when the primary magnetism dies down to zero again. 
When the primary magnetism shifts its north pole back to the 
top, the secondary magnetism will begin to decrease, then fi¬ 
nally build up with its north pole to the right again, and so on. 

The north magnetic poles produced on the motor thus reach 
their maximum in the following order: top, right, bottom, left, 
etc., or in the direction of rotation. Further consideration will 
show that, had the rotation been taken in the opposite direc¬ 
tion, the poles would have traveled in the opposite direction 
also. 

It must be remembered that the simultaneous existence of 
magnetic fluxes at right angles in the same material is entirely 
imaginary. The effect of each is, however, real and the ex¬ 
istence of the resultant is real. 

Production of Revolving Field. 

When the rotor is traveling at synchronous speed, the e.m.f. 
generated in the secondary conductors by their motion across 
the primary magnetism is of such a value as to require the 
induced field (quadrature magnetism) to be equal in effective 
value to the primary field. If the maximum value of the pri¬ 
mary field be taken as unity and time be denoted in angular 
degrees, then the instantaneous value of the primary magnetism 
may be represented by cos. a, where a measures the angle of 
time from the instant of maximum primary magnetism. In 
a similar manner sin. a may represent the instantaneous value 
of the quadrature magnetism. 

These two fields are located mechanically 90 degrees from 


52 


ALTERNATING CURRENT MOTORS. 


each other. Fig. 24 indicates the manner in which the fluxes 
of the two fields vary from instant to instant, and the position 
of the resultant core magnetism at each instant, a two-pole 
motor, with a ring-shaped core without projecting poles, being 
assumed. 

O B = O C cos. a = value and position of primary magnetism 
after the time lapse a. 

O A =OD sin. a = value and position of induced magnetism 
at same instant. 

O C = \/0 B 2 XO A 2 represents the resultant field, both in 
value and position. The point P describes a circle. Without 
further proof it is evident that, neglecting the effect of local 



Fig . 24.—Circular Revolving Field. 


impedance in the secondary circuit, there is produced a re¬ 
volving field of constant intensity when the rotor revolves at 
synchronous speed. 

Elliptical Revolving Field. 

When the rotor speed is not truly synchronous, the extremity 
of the vector 0 P, which represents the value and position of 
the resultant field, describes an ellipse. For any given ef¬ 
fective value of primary field magnetism, the effective value 
of the induced field magnetism depends directly upon the speed 
as mentioned above. Let 5 represent the speed with synchronism 
as unity, then, if cos. a represents the instantaneous value of 
the primary magnetism, SXsin. a equals the instantaneous 
value of the induced magnetism. 














THE SINGLE-PHASE INDUCTION MOTOR. 


53 


Referring now to Fig. 25, after any time lapse, a , 

O B = O C cos. a represents the instantaneous value of the 
_ primary magnetism. 

O A = 5X0 C sin. a represents the value of the secondary 
magnetism, while 

O P = \/0 B 2 +0 A 2 represents the value and position of the 
resultant magnetism. 

Since, from the figure, the distance B P bears a constant ratio 
of 5 to distance B C, the locus of the curve described by the 
point P is an ellipse. 

The vertical axis of this ellipse is determined by the primary 
field, while the horizontal depends upon the speed. Above 



Fig. 25.—Elliptical Revolving Field. 


synchronism the figure remains an ellipse, having its major 
axis along the induced field line. At synchronism the ellipse 
becomes a circle, as noted above. At zero speed the ellipse 
is a straight line, which means that at standstill there is no 
quadrature flux and hence no revolving field. Below zero 
speed, that is, with reversed rotation, the curve is yet an ellipse, 
the side which was previously to the right being transferred 
to the left and vice versa. 

Starting Torque of the Single-Phase Motor. 

« % 

For the above reasons, the single-phase induction motor has 
inherently no starting torque whatever, but will accelerate 
almost to synchronism if given an initial speed m either direc- 










54 


ALTERNATING CURRENT MOTORS. 


tion, and, when the torque is not too great, will operate in a 
manner quite similar to that of a polyphase induction motor. 

Due to the existence of the quadrature flux, to produce which 
magnetomotive force must be supplied by current in the pri¬ 
mary windings, at synchronism the magnetizing current for a 
single-phase motor is twice as great per phase as is the case 
w r hen the same machine is properly wound and operated on a 
two-phase circuit of the same e.m.f.; but the total number of 
exciting ampere-turns is the same in the one case as in the 
other. A difference in the performance of a single-phase from 
that of a polyphase motor is found in the existence at no load 
of considerable current in the secondary with the former, while 
the rotor current is practically negligible with the latter. These 
facts will be discussed more fully in a subsequent chapter. 

Use of “ Shading Coils/’ 

The most serious defect in the behavior of single-phase motors 
is in connection with their lack of starting torque, to remedy 
w r hich many ingenious devices have been developed. A simple 
method of producing the requisite quadrature magnetic flux 
for the purpose of giving a single-phase induction motor a 
starting torque is found in the use of. “ shading coils,” which 
are extensively employed in alternating-current fan motors. 
Each coil consists of a low resistance conductor surrounding a 
portion of a field pole. As ordinarily applied, there is cut in 
each pole a slot parallel to the shaft of the rotor. In this slot 
is placed the conductor, which is connected by a closed path of 
high conductivity around a portion of the pole included between 
the slot and the side of the pole, as shown in Fig. 26. The coil 
of each pole is placed similarly to that of the other poles, and, 
as explained below, the secondary revolves in the direction from 
that portion of a pole not surrounded by a coil towards the 
“ shaded ” side of the pole. 

The action of the shading coils is as follows, reference being 
had to Fig. 26: Consider the-field poles to be energized by 
single-phase current, and assume the current to be flowing in 
a direction to make a north pole at the top. Consider the poles 
to be just at the point of forming. Lines of force will tend to 
pass downward through the shading coil and the remainder 
of the pole. Any change of lines within the shading coil gen- 


THE SINGLE-PHASE INDUCTION MOTOR. 55 

erates an e.m.f., which causes to flow through the coil a current 
of a value depending on the e.m.f. and always in a direction 
to oppose the change of lines. The field flux is, therefore, 
partly shifted to the free portion of the pole, while the accu¬ 
mulation of lines through the shading coil is retarded. How¬ 
ever, so long as the magnetomotive force of the field current 
is of sufficient strength and in the proper direction, the lines 
through the shading coil increase in number, although the 
increase is retarded, which is to say, that, even after the lines 
in the other part of the pole begin to decrease in number, the 



tuO 

•2 —• 
T3 o 
cd o 
.C 



Figs. 26 and 27.—Action of Shading Coils. 


flux within the shading coil is increasing and continues to in¬ 
crease till the exciting current drops to a strength just sufficient 
to maintain that density of flux which is within the coil. At 
this instant the flux in the shading coil has its maximum value 
as a north magnetic pole. 

As the flux on the other portion of the pole continues to de¬ 
crease, the lines within the shading coil tend to decrease also, 
but the e.m.f. generated by their rate of change causes to flow 
a current which tends to prevent any change of lines, so that 
when the other portion of the pole contains no lines whatever 
there is yet within the shading coil an appreciable amount of 































































































56 


ALTERNATING CURRENT MOTORS. 


flux, forming a north magnetic pole, as indicated by Fig. 27, 
the result being a shifting of the field from the unshaded to the 
shaded side of each pole. This is repeated when the mag¬ 
netism reverses. The lines within the shading coil decrease, 
with an increasing rate of change, and a condition of zero 
lines within the coil is soon reached. At this instant, there is 
a south magnetic pole at the “ unshaded ” end of the top 
field pole, and a north magnetic pole at the “ unshaded ” end 
of each adjacent field pole, and south and north magnetic poles 
begin immediately to form within the corresponding shading coils. 

Looking back over the process of formation of the magnetic 
poles, it is seen that north poles have occupied successively 
the following positions: top “ unshaded,” top total, top “shaded,” 
side “ unshaded,” side total, etc. Or, more simply, the north 
pole, though varying in strength, has travelled in a counter 
clock-wise direction. This process is continuous, and results 
in a truly rotating field. A rotor placed in this field is drawn 
into rotation as though the primary had been properly wound 
and connected to an unsymmetrical polyphase circuit. This 
method cannot be satisfactorily and economically applied to 
motors of large sizes. 

Use of Commutator on the Rotor. 

A simple method of applying a commutator to induction 
motors for starting purposes is to utilize current produced in 
the armature by the alternating flux from the field. In the 
application of this method the rotor is provided with a winding 
similar to that of a direct-current armature, connected to a 
commutator in a manner very similar to that commonly em¬ 
ployed with direct-current machinery. Due to facts discussed 
below, current which flows through the armature, by way of 
suitably connected brushes, gives to the rotor sufficient torque 
to bring it under full load to a predetermined speed at which a 
mechanism operated by centrifugal force causes a short-circuiting 
device to inter-connect all the segments of the commutator, and 
thus to convert the armature into virtually a squirrel-cage rotor. 

The function of the commutator in the production of the start¬ 
ing torque may be determined by reference to Fig. 28. Con¬ 
sider a closed coil armature supplied with a commutator to be 
situated at rest between two poles of a motor, which poles 


THE SINGLE-PHASE INDUCTION MOTOR. 


57 


are excited with alternating current, as indicated in the drawing. 
There will be generated in each coil of the armature an alter¬ 
nating e.m.f. of a value depending upon the rate at which that 
coil is cut by the alternating field flux. The coils which lie in a 
horizontal plane will be cut by the maximum flux, while those 
in the vertical plane will be cut by the minimum flux, which 
minimum will be zero when the plane becomes truly vertical. 
Though all the coils are connected in a continuous circuit, no 
current will flow in the armature, since the e.m.f. on one side 



Fig. 28. —Production of Starting Torque. 

is equal to that on the other, and the two sides are in series. 
The maximum e.m.f. will exist between a top and a bottom arma¬ 
ture coil, or between opposite commutator segments, which lie 
in the vertical plane. 

Let E represent the effective value of this maximum e.m.f., 
then E cos. 6 will represent the value of the e.m.f. between 
opposite commutator segments occupying a plane forming an 
angle of 0 degrees with the vertical. 

Consider two opposite commutator segments to be connected 
together externally by a conductor and appropriate brushes, 



































58 


ALTERNATING CURRENT MOTORS. 


and represent the impedance of the armature and the external 
circuit by Z. Then a current will flow through this circuit, 
which current may be represented in value at any given position 
of the brushes by 

E cos. 6 
1 = Z ‘ 

This current will be a maximum when the brushes occupy 
the vertical plane, and a minimum when they are in the hori¬ 
zontal plane. 

Referring again to Fig. 28, and remembering that a conductor 
carrying a current in a magnetic field experiences a torque 
which is proportional to the product of the current, the field 
and the cosine of the angle between them, it wfill be seen that, 
for a given current in the armature, the maximum torque would 
be exerted when the brushes are in the horizontal plane, and 
that the armature will experience no torque whatever when the 
brushes are in the vertical plane. It is to be noted, however, 
that when the brushes are in the horizontal plane no current 
will be produced in the brush circuit, and, therefore, the torque 
is zero. In any intermediate position the current in the brush 
circuit gives a certain torque. 

Obviously, when the current and flux reverse together the 
torque continues to be exerted in one direction and the rotor 
is given the desired initial speed previous to being converted 
to a squirrel cage rotor, as stated above. 

Single-phase motors equipped with starting devices of this 
nature give satisfactory results as to simplicity of operating 
circuits, efficiency of performance and reliability of service 
quite comparable to those obtained with polyphase motors 
This type of machine in its starting condition is frequently 
referred to as a “ repulsion ” motor. The repulsion motor is 
treated at great length both graphically and algebraically in 
subsequent chapters. 

Polyphase Induction Motors Used as Single-Phase Ma¬ 
chines. 

Induction motors are frequently started up from rest on 
single-phase circuits by operating them as so-called “ split-phase” 
machines. Commercially considered, a split-phase motor is a 
polyphase machine, the current in the separate phases being 



THE SIXGLE-PHASE 1XDUCTIOX MOTOR. 


59 


obtained at different lag angles from a single-phase circuit, so 
that a starting torque is produced at the rotor. 

A two-phase induction motor may be brought up to speed 
on a single-phase circuit by connecting the windings of both 
phases to the circuit, one directly and the other through a suit¬ 
ably chosen resistance or condensance, as shown in Fig. 29. 
The current through the circuit containing the resistance will 



Fig. 29.—Circuits of (Two-phase') Fig. 30.—Circuits of (Three-phase^ 
Single-phase Motor. Single-phase Motor. 

alternate more nearly in unison with the impressed e.m.f. than 
that in the other, and it will possess a component in quadrature 
to the current in the other circuit, which component will pro¬ 
duce the desired quadrature flux. The elliptical revolving field 
thus produced will give a torque which will start the motor up 
from rest, if the load be not too great. When about half speed 
has been attained the circuit through the resistance is cut out 
and the motor operates as a single-phase machine. 

Under the conditions of operations, it will be found that there 






































60 


ALTERNATING CURRENT MOTORS. 


is generated in the inactive phase winding an e.m.f. equal (for 
negligible secondary impedance) to the counter e.m.f. of mechan¬ 
ical motion in the active phase winding, and that this e.m.f. 
is almost in quadrature in time-position with the supply e.m f. 
The lack of exact quadrature is due to the lag of the counter 
e.m.f. of rotation in the active coil behind the impressed e.m.f. 
and to some extent to the further lag of the secondary exciting 
current behind this e.m.f., and the sign of the angle of dis- 



e.m.f.’s, e.m.f.’s. 

placement from 90° depends upon the direction of rotation of 
the motor secondary. Fig. 31 shows the relative value and 
position of this tertiary e.m.f. for a two-phase motor running 
single-phase, while Fig. 32 indicates equivalent results for a 
three-phase motor on a single-phase circuit. 

By combining the e.m.f. of the inactive winding of a two- 
phase motor with that of the supply circuit, there is available 
an almost symmetrical two-phase circuit, from which may be 
started at once, without auxiliary apparatus, any similar two- 
phase, or, by a few slight changes, any three-phase induction 










THE SINGLE-PHASE INDUCTION MOTOR . 


61 


motor. The draught of current from the inactive phase winding 
will have very little effect upon the operation of the first motor. 
By this method it is possible to dispense with auxiliary starting 
apparatus for all motors of any given installation, with the ex¬ 
ception of one, and, with properly arranged circuits, two-phase 
motors may in this manner be started up with a fair operating 
torque. 

A three-phase induction motor may be operated from a single¬ 
phase circuit by connecting two leads from the motor directly 
to the supply circuit and joining the third to an auxiliary 
starting circuit, formed by placing a resistance and a reactance 
in series across the supply circuit, as indicated by Fig. 30. 

The effect of placing the resistance and reactance in series 


c 



Figs, 33 and 34. —Vector Diagram of Electromotive Forces. 

is to displace the relative potential of the point where the two 
join, from a line connecting the extremities of the two, as shown 
in Fig. 33. For a true reactance the locus of this point, with 
varying resistance, is the arc of a circle, as indicated by Fig. 34, 
and the maximum displacement occurs when the resistance 
and reactance are equal. Under this condition the displacement 
(when no current is being taken off at P ) will be .5 E Tl or one- 
half the line voltage. For a true three-phase circuit, the dis¬ 
placement should be 

V3 

~2~ E t — .866 E t . 

It is clear, therefore, that this method cannot possibly give e.m.fs. 
in true three-phase relation. It is found, however, that the 










62 


ALTERNATING CURRENT MOTORS. 


displacement obtained is adequate for starting motors of mod' 
erate sizes. 

The use of condensance, instead of inductance, offers some 
advantages, since by properly proportioning the condensance, 
the leading current demanded may be adjusted to equality 
with the lagging exciting current during operation and, theoret¬ 
ically, a power factor of unity may be obtained. The disturb¬ 
ing influence of change of frequency, .the compensating dis¬ 
advantages due to presence of higher harmonics from the 
distortion of the e.m.f. wave from a true sine curve of time- 
value, and the practical necessity of operating condensers at 
high voltage, coupled with the lack of satisfactory commercial 
condensers in convenient form, have limited the application of 
this method. 


CHAPTER VI. 


GRAPHICAL TREATMENT OF INDUCTION MOTOR PHENOMENA. 

Advantage of Graphical Methods. 

With almost no exception, the graphical method of treatment 
of electrical phenomena does not produce results as accurate 
as the analytical; yet, for many purposes where a fine degree 
of accuracy is not required, the ease of manipulating has led to 
the extensive use of the graphical method, approximate results 
being first rapidly determined, after which if greater accuracy 
is desired, the analytical method may be used. In cases where 
only qualitative results are desired, but where some knowledge 
of the effect of change in the different variables connected with 
the phenomena is important, the graphical method, because of 
its simplicity, readily lends itself to the quick determination 
of results sufficiently accurate for the needs of the case. 

Before discussing the graphical diagram for the re present a"-’'on 
of the value and phase of primary and secondary currency of 
an induction motor, it is well to establish the similarity between 
an induction motor and a static transformer. 

Effect of Inserting Resistance in the Secondary. 

At any value of field magnetism, the effect of inserting re¬ 
sistance in the secondary of an induction motor, for a given 
value of secondary current, is to vary the slip directly with the 
total secondary resistance without affecting either the torque 
or the power-factor. 

As proof of this fact, let E 2 = the e.m.f. which would be 
generated in the secondary at 100 per cent, slip, at the given 
field magnetism. 

5 = slip, with synchronism as unity, 

A, = secondary resistance, 

X 2 = secondary reactance at 100 per cent, slip (standstill;5 = 1) 
/ 2 = secondary current. 


63 


64 


ALTERNATING CURRENT MOTORS. 


Then 


j = sE 2 _ 

2 VR 2 2 +s*X 2 2 


or I 2 = - 7 


5 2 £ 2 2 


R 2 + s 2 X* 


or, since I 2 and E 2 are constant for the chosen condition of service, 


K 2 = 


R 2 + s 2 X 2 


52 (l -K 2 X 2 )^ a%R ^ 

or the slip is directly proportional to the secondary resistance. 
The secondary power-factor, cos. 0 — 

R 2 

VR 2 2 + s 2 X 2 2 = 

r 2 = _ 1 _ 

\/R 2 V\+a 2 X 2 Vl + a 2 X 2 


or the secondary power-factor is independent of the secondary 
resistance. 

The total secondary power is / 2 E 2 cos. 6, and is independent 
of the secondary resistance, while the torque, which is 


I E p 0 

7.04 —-— 2 2 r, is also independent of the secondary resistance. 
syn. speed 

Since the effect of inserting resistance in the secondary cir¬ 
cuit is merely to increase the slip without altering the other 
quantities, it follows that by using suitable selected resistances, 
the slip can be made unity for any value of secondary current 
at the corresponding power-factor. In consequence of this 
fact, the performance of the secondary will be faithfully repre¬ 
sented if all the power received from the primary be considered 
as dissipated in resistance in the secondary circuit, the slip at 
all times being taken as unity and the total secondary resist¬ 
ance (conductance) being assumed to be varied according to 
the secondary load, or briefly, the induction motor may be 
treated in all respects like a stationary transformer. The 
determination of the slip, torque, etc., of the motor under oper¬ 
ating conditions will be discussed later. 
















TREATMENT OF INDUCTION MOTOR PHENOMENA. 65 


Primary and Secondary Current Locus. 

Perhaps, of all the graphical diagrams which have been sug¬ 
gested at various times to represent the performance of an 
induction motor, the simplest, and at the same time the most 
complete, is that showing the value and phase of the .primary 
and secondary currents. The quantities intended to be repre¬ 
sented by this diagram can best be ascertained by investigating 
the method of determining the points on the current locus, 
such as is shown in Fig.-35, which is plotted according to the 
following instructions: 

Use the vertical scale (at a certain number of amperes per 



Fig. 35. —Primary and Secondary Current Locus. 


inch) to plot values of the power component of the currents 
(/ cos. 0)] and the horizontal scale (at the same number of am¬ 
peres per inch) to plot the wattless component of the currents 
(/ sin 0). 

From the origin O, lay off a distance O B equal to the power 
component of the primary current at no load, and from the 
point B draw the horizontal line B A with a value equal to 
that of the wattless component of the primary current at no 
load. The line 0 A represents the no load primary current, 
while the angle A OF is the primary angle of lag at no load. 
In a similar manner, selecting any load current, as 0 0, lay off 































































6G 


ALTERNATING CURRENT MOTORS. 


PR 

OR 

POR 

PQ 


11 


the power and wattless components O TO and D C. The angle 
of lag at this load is represented by the angle C 0 P, Follow¬ 
ing out the same method, locate a number of points corre¬ 
sponding to the points .4 and C, and draw a smooth curve 
through them. At any point on this curve, as P, 

0 P represents the primary current, 

wattless component of the primary current, 
power component of the primary current, 
primary angle of lag. 

wattless component of the secondary cur¬ 
rent. 

A Q “ “ power component of the secondary current 

A P “ “ secondary current, 

P A Q “ “ secondary angle of lag. 

The proof of the representation of the secondary quantities 
is as follows: When running under load, B R equals the increase 
in the power component of the primary current over its no load 
value. As in a transformer, this increase is due to the flow 
of current in the secondary, and being in phase (opposition) 
to the power component of the secondary current, is a direct 
measure of its value. A similar course of reasoning holds for 
the wattless component P Q. Hence A P represents the sec¬ 
ondary current both in value and phase position. 

An inspection of the diagram will show that the maximum 
secondary power factor occurs at no load, while the maximum 
primary power factor occurs at that load which causes the 
line O P to become tangent to the curve A C P F. Since the 
no load losses are equal to the product of 0 B by the primary 
e.m.f., and the secondary current, primary current and power 
factor can be obtained directly from the curve for any value of 
primary load, it follows that if the primary and secondary re¬ 
sistances be known, the input losses, output, efficiency, slip, 
speed, power-factor, apparent efficiency, and torque can easily 
be determined when the curve A C P F is located. 


Test Results. 

The accompanying table records results of calculations made for 
a 10-h.p., two-phase, 220-volt induction motor, use being made of 
Fig. 35, which has been constructed in part from data obtained 
with this machine. Results found in the table have been plotted 


TREATMENT OF INDUCTION MOTOR PHENOMENA. 67 



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68 


ALTERNATING CURRENT MOTORS. 


in the form of the curves shown in Figs. 36 and 37, from which 
can be ascertained the complete performance of the motor 
from no load to full load and beyond. This motor is supplied 
with a variable resistance external to the secondary windings 





01234567 89 

Horse Power Output 



Figs. 36 and 37. —Performance of Two-phase Induction Motor. 


for the purpose of decreasing the primary current and increasing 
the rotor torque during the starting period. The effect of 
using this resistance is seen at a glance when Fig. 36 is com¬ 
pared with Fig. 37. 






























































































































































TREATMENT OF INDUCTION MOTOR PHENOMENA. .69 


From column 9 of the table it is seen that the maximum torque 
of the rotor occurs at that value of the primary current which 
allows the greatest amount of power to be delivered to the 
secondary and that the value of this torque and the primary 
current and power-factor at which it occurs are entirely indepen¬ 
dent of the secondary resistance. Columns 10 and 11 of the 
same table, however, show that the slip at which occurs the 
maximum torque is directly proportional to the secondary re¬ 
sistance, though the secondary current which gives this torque 
has a definite fixed value independent of both the slip and the 
secondary resistance. It is evident, therefore, that by the use 
of variable resistance in the secondary circuit, the maximum 
torque can be made to occur at any speed from standstill to a 
few per cent, below synchronism, but that the primary current 
will depend upon the torque and not upon the speed. 

While the method of calculation used in arriving at the re¬ 
sults given in each column of the table is explained at the heads 
of the separate columns, perhaps a few words should be added 
concerning the formulas for determining the slip and the torque. 
The rotor slip may be expressed as the ratio of the copper loss 
of the secondary to the total power received from the primary. 
That is, 


where W s is the total secondary power. This relation was dis¬ 
cussed at length in a previous chapter, and it need not be 
dwelt upon at this place. Likewise it has been shown that 
the rotor torque may be expressed in pounds at one foot radius 
by the formula, 

D 7.04 W s 
syn. speed’ 

It will be observed from column 7, that there has been 
added to the primary copper loss a quantity, 856 watts, to obtain, 
the total primary loss. The 856 watts is the so-called con¬ 
stant loss,” and includes all iron and friction losses of the 
motor. It should be carefully noted in this connection that 
this loss is not constant, but it varies with increase of load. 
There are three causes which affect the constancy of the value of 
the iron loss. The drop in e.m.f. due to the current through the 




70 


ALTERNATING CURRENT MOTORS. 


primary resistance lessens the e.m.f. to be balanced by the rate 
of change of the core flux and thus decreases the density of mag¬ 
netism and tends, thereby, to reduce the iron loss. When the rotor 
travels at synchronous speed the secondary core experiences no 
reversal in magnetism and there is, therefore, no secondary iron 
loss. As load is placed upon the motor, the rotor speed de¬ 
creases and the magnetism of the secondary core reverses at a 
rate proportional to the slip and there is produced a correspond¬ 
ing core loss, tending to increase the total iron loss of the motor. 
Another cause tending to increase the iron loss of both the 
primary and secondary cores is the loss in the iron immediately 
around each conductor due to the superposed local flux when 
current traverses the conductor. The flux causing this local 
loss is that to which is due the reactance of the primary and 
secondary coils. The value of this flux depends directly upon 
the current in the conductors and inversely upon the total 
reluctance of the magnetic path surrounding them. The loss 
due to a given flux in a certain mass of iron depends not only 
upon the value of the flux and the frequency of its reversal 
but to a large extent also upon the density of magnetism upon 
which this flux is superposed. In any commercial motor the 
density of magnetism in the core teeth is normally quite high 
on account of the field magnetism proper alone, so that the loss 
due to the local flux surrounding each slot in both the primary 
and secondary cores under load currents depends greatly upon 
other factors than its own value and periodicity of reversal. 
The increase of iron loss due to the second and third causes, 
as enumerated above, is ordinarily somewhat greater than the 
decrease due to the first cause, resulting in the so-called load 
losses. In comparison with the total losses the increase in iron 
loss is usually quite small throughout the operating range of 
the motor. When the character of the work necessitates such 
procedure, this increase can be approximately determined and 
corresponding corrections made, though other errors in com¬ 
putations or assumptions will frequently more than equal that 
due to the neglect of this increase. 

Equation of the Current Locus. 

Referring now to Fig. 35, it will be seen that the curve A C P F, 
the current locus, has been drawn as the arc of a circle. This 


TREATMENT OF INDUCTION MOTOR PHENOMENA. 71 


is the construction usually adopted for the purposes of pre¬ 
liminary design, in which case the point F is located as the 
extremity of the vector representing the value and phase of the 
primary current, which would be obtained with full e.m.f. 
impressed upon the motor with the rotor stationary. This 
construction is partly justified by the following fact: 

If the primary resistance and the exciting current be neglected, 
or the secondary current be considered equal to the primary, 
the current locus is a true circle. 

Let E = impressed e.m.f., 

I = current to the motor, 

X r = primary reactance, 

X 2 = secondary reactance, 

R 2 = secondary resistance. 

Treating the induction motor as a transformer, the value of 
the current as Ro is varied is 

/- - £ 

V {X x +X 2 f + R 2 2 

but R 2 = (Xy+X 2 ) cotan d; hence I = 

E E . 

-7- - ; - = -=7— , - T y - SM 0 

( X t + X 2 ) \/1 + cotan 2 0 ^1 + ^2 

which, when E is constant, is the polar equation of a circle 
having a diameter of Zs-f- (Xj + X 2 ). 

Errors in Assuming a Circular Arc. 

If the primary coils carried at all times an exciting current of 
constant value in addition to a current equal and opposite to 
the secondary current, the curve would yet be a true circle. 
In this case, however, the primary current would need to be 
measured from a point external to the circle at a distance from 
the circumference equal to the exciting current, that is, the 
primary current would be measured from B while the secondary 
current would, as formerly, be measured from A. 

If in addition to the exciting current the primary coils carried 
a power component of current for the no-load iron losses, etc., 
over and above the counter current to oppose the current in 
the secondary, the locus would remain a circle, as before, but 
the primary current would be measured from a point 0 so 
located that 0 B equals the power component of the no-load 







72 


ALTERNATING CURRENT MOTORS. 


current, Fig. 35. The last assumption is true for any induction 
motor in which the drop in e.m.f. due to the passage of the 
current through the local impedance of the primary coils is 
negligible. The local primary impedance is, however, not 
negligible in a commercial motor. It is found, however, that the 
error in determining the primary and secondary currents due to 
the slight deviation of the curve from a true circle between the 
points A and 5 produces an almost inappreciable effect upon the 
results. 

The effect of the primary impedance is the same in all re¬ 
spects as though the primary coils were without impedance and 
the power supplied to the motor were transmitted over a circuit 
having an impedance equal to that of the primary, which means 
that in the equation, E decreases with an increase of /, and 
the equation is, therefore, not that of a true circle. In a sub¬ 
sequent chapter these facts will be discussed more fully. 

Having drawn attention to these errors in the assumptions 
concerning the current locus, it is sufficient here to state that 
while the method used does not give the absolutely true loca¬ 
tion of the primary current curve throughout its whole length, 
and the absolutely true secondary current curve does not 
coincide with that of the primary, the errors introduced into 
the calculations are relatively small and for most practical 
purposes may well be neglected. 

The product of the power component of the primary current 
by the circuit e.m.f. gives the input to the motor. It will be 
seen from Fig. 35 that the maximum power which it is possible 
for the motor to receive is determined by the radius of the 

E 2 

circle A PF, that is, it is represented by the quantitv-^- 

2(X 1 + X 2 y 

(neglecting the no-load losses), or the maximum power which 
the motor can receive is determined wholly by the reactance 
of the primary and secondary coils. 

A further inspection of Fig. 35 will show that the maximum 
power factor occurs at that value of primary current which 
causes the line OP to become tangent to the curve A PF. 
The value of this power factor depends upon the radius, 

E 


2(A 1 + X 2 ) 




TREATMENT OF INDUCTION MOTOR PHENOMENA 73 


of the arc A P F and upon the exciting current A B, being 
increased as the former is increased or as the latter is decreased. 

Effect of Design on LEAKAbE Reactance. 

It is evident from the foregoing that in the construction of 
induction motors every effort should be made to render the 
reactance of the coils as small as possible. The reactance flux 
can be decreased by increasing the reluctance of the path 
which it must travel, that is by using open slots and operating 
the core teeth at high magnetic density. A method of de¬ 
creasing the effectiveness of the local flux without, however, 
lessening appreciably the number of lines is found in the use of 
many rather than few slots. Thus when the conductors are 
bunched in a single slot each conductor is cut by the lines 
due both to its own current and to that of its neighbors in 
the same slot, so that the reactive e.m.f. per conductor depends 
directly upon the number of conductors in each slot, or the 
reactive e.m.f. per slot varies with the square of the conductors 
therein, and the total reactance of a certain winding decreases 
with an increase in the number of slots in which the conductors 
are placed. 

The value of the exciting watts depends almost entirely 
upcn the radial depth of the air-gap (more properly, upon the 
volume of the air-gap) and the employment of a small air-gap 
is desirable for large operating power factor. An inspection 
of Fig 35 will, however, reveal the interesting fact that since 
the radius of the arc A P F is independent of the distance 
A B, the maximum power of the motor is unaffected by the 
value of the air-gap except as a change in the latter may affect 
the reactance of the coils. Since a reduction in the air-gap 
is accompanied with an increase in the local reactance flux 
around the coils, one is led to the highly interesting conclusion 
that reducing the air-gap of a given motor actually decreases 
the maximum power of the machine. 

These facts are discussed more fully in a subsequent chapter, 
while numerical valves for the leakage reactance of various 
designs are given in the appendix. 


CHAPTER VII. 


INDUCTION MOTORS AS ASYNCHRONOUS GENERATORS. 

Operation Below Synchronism. 

The current demanded from the supply system by an in¬ 
duction motor when operating without load near synchronism 
is found to consist of two components, one in phase, and the 
other in quadrature with the impressed electromotive force. 
The inphase, power component is found as that value of amperes 
which multiplied by the impressed volts will give the watts 
necessary to supply the internal losses of the machine, while 
the quadrature component is found as that value of amperes 
which multiplied by the number of turns of the primary wind¬ 
ing will give the magnetomotive force necessary to cause to 
flow through the reluctance of their paths the lines required 
to produce by their rate of change an internal counter 
e.m.f. less than the impressed by an amount such as to 
allow the primary current to flow through the impedance of 
the coil. 

When the speed is less than synchronism, the secondary 
windings cut the revolving field at a rate proportioned to the 
slip and the e.m.f. thus generated causes to flow through the 
secondary conductors a current, which being practically in 
time-phase with the magnetism and in mechanically the same 
position in space, will give to the rotor a torque in a direction 
to lessen the secondary current, that is, a torque tending to 
accelerate the rotor. This current in its effect upon the pri¬ 
mary acts as though it flowed in the secondary circuit of a 
stationary transformer and thus requires in the primary coil a 
current equal and opposite to it in magnetomotive force, so 
that the core magnetism is but slightly altered by its presence 
The counter current appears as an addition to the power com¬ 
ponent of the primary current and lepresents the increase in 
electrical power supplied to the motor. 

74 


INDUCTION MOTORS AS GENERATORS. 


75 


Operation Above Synchronism. 

When the rotor is driven above synchronism, the windings 
of the secondary cut the synchronously moving magnetism in 
a direction to generate an e.m.f. causing a current to flow in the 
secondary conductors in a direction opposite to that described 
above, so that the additional component of current required 
in the primary coil is reversed from its former time-phase 
position with respect to the field magnetism and to the im¬ 
pressed primary e.m.f., and thus tends to represent power 
flowing from the motor and, as will be discussed in detail later, 
at a speed sufficiently far above synchronism, an induction motor 
acts as a generator, the power delivered by it depending upon 
the relative motion of the secondary windings and the field 
magnetism, that is to say, upon the slip above synchronism. 
This characteristic of the machine has been frequently availed 
of upon mountain roads, where, in descending, the primary 
circuit of the motor is connected directly to the trolley line 
and the speed of the rotor held practically constant at a few 
per cent, above synchronism, or increased to any desired value 
by the insertion of resistance in the secondary circuit. Where 
the system of tandem speed control of two similar motors is 
used, a breaking and energy-restoring effect may be produced, 
even upon level roads, by placing the motors in tandem con¬ 
nection while they are yet traveling at a speed between one- 
half and full synchronism, and the rate of retardation may be 
rendered quite uniform by judicious use of the starting resist¬ 
ance. The same remarks obviously apply when the speed con¬ 
trol is obtained by change in the number of magnetic poles of 
a single motor. 

Vector Diagram of Currents. 

The relative values and phase positions of the primary and 
secondary currents of an induction motor at all speeds both 
below and above synchronism may be graphically represented 
by a simple diagram which lends itself to a ready interpretation 
of the interdependance of the various characteristics of such a 
machine. Fig. 38 gives a vector diagram of the primary cur¬ 
rent of a certain three-phase, asynchronous machine operating 
under a constant impressed e.m.f. of 220 volts, and from this 
diagram may be determined the secondary current and all of 


70 


ALTERNATING CURRENT MOTORS. 


the performance characteristics of the machine and thus may 
its behavior as an asynchronous generator conveniently be in¬ 
vestigated. In Fig. 38 the distance 0 B represents the power 
component of the primary current at no-load, the distance A B 
is the wattless component of the no-load current, while O A 
represents both in value and phase position the total current 

Wattless Component of Current 



Fig. 38.—Current Locus of Asynchronous Machine. 

taken by the motor at no-load. The product of O B with the 
impressed e.m.f. gives the no-load losses of the motor, while 
the ratio of OB to O A, the cosine of the angle A O B, is the 
primary power-factor under no-load conditions. At a certain 
slip, the primary current will in crease to some value and phase 
such as is represented by 0 P, of which O R is the power, and 
R P, the reactive component respectively. 
























































INDUCTION MOTORS AS GENERATORS. 


77 


Now, the increase in the two components of the primary cur¬ 
rent under load is due to the corresponding components of the 
secondary current, so that such increase serves as a measure of 
the secondary current. An inspection of Fig. 38 will show that 
the line A P is the vector sum of the changes in the two com¬ 
ponents of the primary current over the no-load values and 
hence represents both the value and phase (opposition) position 
of the secondary current when reduced to primary terms by 
the inverse ratio of turns of the respective windings. A further 
inspection and study of Fig. 38 will show that a series of points, 
such as P could be located for corresponding values of the pri¬ 
mary current and that the locus of such point would form a 
continuous curve representing the value and time-phase position 
of the primary and secondary currents throughout the operating 
range of the motor. 

On account of the fact that, independent of the method by 
w T hich the e.m.f. may be produced in the secondary circuit of 
the motor, the primary circuit is that of a static transformer, 
the locus of the primary current as here designated approxim ates 
closely a circle, the center of which is located on the line B A 
prolonged and the diameter of which is equal to the ratio of the 
impressed e.m.f. to the combined local leakage reactance of the 
two coils, as is true with any stationary transformer. The posi¬ 
tion of the complete circle is determined and it may readily be 
drawn both to the right and to the left of the origin of vectors, 

when the initial point, A, and any load point, P, are located, 
as was discussed in the preceding chapter. 

From the current locus of Fig. 38, the components of any 
chosen value of primary current and the corresponding secondary 
current may be ascertained at once and when the resistances of 
the primary and secondary coils are known, the complete per¬ 
formance of the machine may be determined by simple calcula¬ 
tions. Such calculations are recorded in the table, and the 
results thereof are represented graphically in Fig. 39. The 
methods employed in obtaining the results sought will be ap¬ 
preciated from a study of the head-lines of the several columns 
of the table. 

In all cases, the current referred to is the “ equivalent single¬ 
phase current ” or the corresponding component thereof. This 
term is used to express that value of current which multiplied 





CALCULATION OF PERFORMANCE OF THREE-PHASE ASYNCHRONOUS MACHINE. 

220 Volts, 6 Poles, GO Cycles. 


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INDUCTION MOTORS AS GENERATORS. 


79 


by the circuit e.m.f. and power-factor will give the true 
watts; and the use of such term greatly facilitates any calcula¬ 
tions with polyphase quantities, since all quantities may then 
be treated as though of single-phase significance. 

Similarly, the resistance used is the equivalent single-phase 
quantity, having that value which multiplied by the square of 
the equivalent single-phase current will give the true copper 
loss of the circuit considered. It is an interesting fact, previ¬ 
ously verified, that, for any given two- or three-phase circuit, how¬ 
soever inter-connected, the equivalent single-phase resistance is 



Fig. 39.—Test of Asynchronous Machine as Motor and as 

Generator. 


just one-half of the value found between leads by means of direct 
current measuring instruments. 

Performance Characteristics. 

The curves of Fig. 39 show the performance of the asynchron¬ 
ous machine when its speed is varied from 85 per cent, con¬ 
tinuously to 115 per cent, of synchronism and are the charac¬ 
teristic curves which would be experimentally obtained by 
belting the asynchronous induction machine to a shunt-wound, 
direct-current machine which being supplied with constant 
e.m.f. could be driven throughout this range of speed by varia- 





















































































































































































































80 


ALTERNATING CURRENT MOTORS. 


tion of its field strength, thereby converting it from a generator 
to a motor gradually, as desired. 

As seen, from Fig. 39, at synchronous speed the induction 
motor receives all of its power electrically from the supply 
system and delivers no mechanical power. At 106 per cent, of 
synchronism this particular machine receives all of its power 
mechanically and delivers no electrical power whatever. Be¬ 
tween these two speeds, all power received by the machine is 
dissipated in internal losses and the demand for power by the 
machine is gradually, with the negative slip, transferred from 
the electrical to the mechanical source of supply. Below syn¬ 
chronism, the machine gives out mechanical power, while above 
106 per cent, of synchronism the power given out is electrical; 
that is, the machine operates as a generator and returns power 
to the electrical supply system. 

Such an ^asynchronous generator can be connected directly to 
the supply network without the necessity of first bringing the 
machine to the exact speed corresponding to the circuit fre¬ 
quency, and the portion of the load which it will assume can be 
adjusted quite accurately by variation of the speed of its prime 
mover. 

Paralell Operation of Asynchronous Generators. 

An asynchronous generator, as here assumed, will operate 
satisfactorily in parallel with generators of the synchronous 
type—the ordinary alternators; the frequency of the current 
delivered by it, however, is in any case less than that corre¬ 
sponding to the speed of its rotor, the difference being due to 
the requisite motion of the secondary conductors with reference 
to the primary field. The division of the load between synchron¬ 
ous generators working in parallel is determined by the relative 
phase position of the machines; the generator which maintains 
a phase position in advance of others carrying the greatest 
load, while one which lags behind in phase position is more 
lightly loaded. The actual speed of all the generators thus con¬ 
nected, however, must be the same; it is merely the tendency 
to different speeds, as fixed by the governing mechanism of 
the prime movers, which determines the load division. As far 
as concerns the prime movers, the condition of parallel operation 
of asynchronous generators is quite similar to the above, but 


INDUCTION MOTORS AS GENERATORS. 


81 


in the latter case the load division is determined by the actual 
operating speed of each individual machine; that is, the load 
carried by each is determined wholly by the variation of the 
speed of its rotor from that corresponding to the circuit fre- 
quencv. When its speed is below the circuit frequency, an 
asynchronous generator is driven as an induction motor, which 
fact allows an asynchronous generator to be connected to the 
operating circuit without being brought to exact synchronous 

speed. 

Excitation of Asynchronous Generators. 

As is true with the synchronous alternator, the asynchronous 
generator is not inherently self-exciting but current for excita¬ 
tion must be supplied from some source external to itself. 
Similarly, if the delivered (or supplied) e.m.f. is to be kept con¬ 
stant, the exciting current must be increased as the load in¬ 
creases. When the asynchronous generator is delivering power 
to constant potential mains, the exciting current automatically 
adjusts itself to correspond to the demands of the load. This 
characteristic of the machine will be appreciated from a study 
of the curve of Fig. 39, marked “ wattless current.” Such 
curve, when plotted to the proper coordinates, resembles closely 
the curve representing the relation between external load am¬ 
peres and internal field current of the synchronous alternator. 

A further analysis of the various components of the currents 
reveals the fact that, when operating as either a motor or a. 
generator, the asynchronous machine possesses definite load- 
speed characteristics which are unalterable by any condition 
external to the machine. Thus, at any certain speed, the 
machine demands from the network a definite value of wattless 
current and delivers a certain amount of power either electrical 
or mechanical independent of the requirements of other machines 
connected to the system. As a generator, the machine can 
deliver no wattless current whatsoever and its own supply of 
such current must be derived from some other source. 

When an asynchronous generator and a synchronous motor 
or converter are connected simultaneously to the supply sy s- 
tem, the field excitation of the synchronous machine may be so 
adjusted as to cause this machine to supply the amount of 
wattless current demanded by the asynchronous generator, 


82 


ALTERNATING CURRENT MOTORS. 


while the speed of the asynchronous generator may be so regu¬ 
lated that it supplies just that amount of power current required 
for the synchronous machine. Under these conditions, the 
machines so interchange currents that they, considered as a 
unit, may be disconnected from the network without affecting 
the operation of either machine. The e.m.f. of the set may be 
adjusted as desired by variation of the field strength of the 
synchronous machine. The inherent regulation of the e.m.f. 
of the set, as the load thereon is varied, depends upon the mag¬ 
netic circuits of the two machines and, in general, the per¬ 
formance is stable only when at least one of the machines is 
operated under conditions approximating magnetic saturation. 

With circuits arranged as in Fig. 40, it may be shown experi¬ 
mentally that, under any given condition of operation, all 



Asynchronous 

Generator 

Fig. 40.— Arrangement of Load and Exciter Circuits; Syn¬ 
chronous Converter Excitation. 


wattless current comes from the synchronous machine while 
all power current comes from the asynchronous generator. 
The synchronous converter may supply power electrically from 
its commutator, mechanically from its shaft, or power may be 
derived directly from the mains at the generator. The slip of 
the rotor from the speed corresponding to the circuit frequencv 
will depend upon the amount of power thus demanded, so that 
with constant rotor speed, the circuit frequency will vary with 
the load on the set. The speed of the synchronous machine 
will \ ar\ with the circuit frequency, so that, independent of 
the effect of any wattless current which may be demanded, 
operation at constant circuit e.m.f. can be obtained only when 

the field strength of the synchronous machine is varied with 
the load. 







































INDUCTION MOTORS AS GENERATORS. 


83 


Any demand for lagging current from the set necessitates an 
increase in the lagging component of current from the syn¬ 
chronous motor and weakens the field strength of this machine 
and thereby lowers the circuit e.m.f. A demand for leading 
current produces the opposite effect, and, if the leading current 
be properly adjusted in value, no wattless current whatever 
will flow from the synchronous machine and it may be discon¬ 
nected from the circuit without affecting the performance of 
the asynchronous generator. Ah additional synchronous motor 
with its field cores over-excited may be used as a source of leading 
current, or static condensance with proper means for adjust¬ 
ment may be employed for this purpose. 

Condensers as a Source of Exciting Current. 

If the source of excitation be a synchronous alternator, the 
effect of the presence of the exciting current for the asynchronous 



Fig. 41. —Asynchronous Generator; Condenser Excitation. 

machine may be eliminated by placing across the generator 
circuit condensers adjusted in capacity so as to take an amount 
of leading current equal to the lagging current of the asyn¬ 
chronous generator. (See Fig. 41.) If this adjustment be 
exact, and there be no fluctuations in the load, no current will 
pass from the synchronous to the asynchronous machine, and 
the circuit between them may be opened without producing any 
effect whatever upon tne operation of either machine. Under 
the conditions here assumed the asynchronous generator will 
receive its exciting current from the condensers, and, so long 
as the load remains constant and no changes occur in the speed 
of the prime mover, it will automatically maintain its e.m.f. 
at a constant value. If additional load be now placed upon 
the generator, the e.m.f. will decrease, while, if the load be 
decreased or removed entirely, the e.m.f. will at once increase 














84 


ALTERNATING CURRENT MOTORS. 


the performance being similar in many respects to that of a 
direct-current shunt-wound generator. 

It is believed that the condenser excitation of asynchronous 
generators offers sufficiently varied application of the charac¬ 
teristics of the apparatus involved to justify an extended dis¬ 
cussion of its use. 

Before passing to a discussion of the performance of an 
asynchronous generator when excited by static condensance 
it may be well to recall a few of the fundamental facts con¬ 
nected with the operation of condensers in alternating current 
circuits. 

Condensers in Alternating-Current Circuits. 

When two conducting bodies in close proximity are connected 
to opposite terminals from a source of electrical energy, it is 
found that there accumulates upon the adjacent surfaces or 
within the intervening insulating material, i.e., the dielectric, 
a condition of sub-atomic activity termed “ electricity.” Other 
conditions remaining the same, the amount of electricity, or 
the charge which the bodies w T ill store under a certain potential 
difference varies inversely as the distance separating the plates, 
and is greatly affected by the character of the dielectric. An 
assembly of numerous conducting plates separated by sheets 
of dielectric material as thin as practicable and yet of sufficient 
thickness to give the requisite insulating strength to withstand 
the maximum e.m.f. to be impressed at the terminals, forms the 
essential features of the commercial condenser. 

In order now to investigate the action of a given condenser, 
assume one connected to a source of electrical energy, of which 
the e.m.f. may be changed at will, Assume further that at the 
moment of connecting the condenser in circuit, the e.m.f. is 
of zero value but increasing in a positive direction; charge will 
pass into the condenser, tending to produce, by strain in the 
dielectric, a counter e.m.f. equal to the impressed. Evidently 
so long as the e.m.f. continues to increase, charge will continue 
to pass into the condenser at a rate proportional to the in¬ 
stantaneous increase in e.m.f. When the e.m.f. reaches its 
maximum value, the condenser will have received its full charge 
and no additional amount of electricity will flow thereto. As 
the e.m.f. decreases, charge will flow from the condenser tending 


INDUCTION MOTORS AS GENERATORS. 


85 


at each instant to make the amount remaining therein correspond 
to the instantaneous value of e.m.f. From the above facts it 
will be^seen that the rate of transfer of charge, i.e., the current 
in the circuit, will have a value represented by the rate of 
change of the e.m.f. and if the e.m.f. follows a sine curve of 
time-value the current will follow the corresponding cosine curve, 
and, as seen above, wall be 90 times degrees ahead of the e.m.f. 

For purpose of comparison of the performance of different 
condensers, it is convenient to specify the amount of charge 
which a given condenser will assume under a certain e.m.f. 
relative to some charge taken as a unit. If a certain con¬ 
denser, when subjected to a continuous pressure of one volt, 
accumulates electricity to the amount of one coulomb, that is 
the amount of electricity represented by one ampere for one 
second, it is said to possess a capacity of one farad. A con¬ 
denser of capacity C is one which under a continuous electro¬ 
motive force of one volt assumes a charge of C coulombs or 
when the e.m.f. is E volts the charge will be E C coulombs. 

When the e.m.f. changes there is a flow of current to or from 
the condenser so that at each instant the charge in the con¬ 
denser tends to adjust itself to correspond with the instanta¬ 
neous value of the e.m.f. If the e.m.f. changes at a constant 
rate of one volt per second, then the charge must vary C coulombs 
per second, or the current must have a steady value of C am¬ 
peres, or, in general, the current must be C times the rate of 
change of the e.m.f., that is 

. ~de 
t=c it 

now e = E m sin co t 

where E m is maximum e.m.f. and oj, electrical 

angular velocity, 

de 

hence -r = oj E m Cos. oj t so that 
dt 

i = C oj E m Cos. co t or the maximum current, I m = C oj E m , 
and, virtual values being used throughout, 

I = C oj E 

which means that if a condenser of capacity C be connected to 


86 


ALTERNATING CURRENT MOTORS. 


a source of alternating e.m.f. E when the frequency is f there 
will flow therein an alternating current of value I such that 

I ==CE(o^=CE2r[j. 

In alternating current circuits, the ratio E to I gives the 
impedance which for a condenser as above is 


I CojE Coj 

to which specific quantity there is given the name condensance, 
or negative reactance. 

In the statement above concerning the angle of lead of the 
current taken by a condenser, a fact of minor importance has 
been neglected. It is found that the value of energy which a 
condenser returns upon discharge is less than that received 
during charge, the difference being dissipated in heat loss within 
the condenser through dielectric hysteresis and to a small 
extent to the heating effect of the current upon the conducting 
material. This loss requires a component of current in phase 
with the e.m.f. across the condenser terminals, so that in any 
practical condenser the current leads the applied e.m.f. by a 
time-angle less than 90°. The energy component of the con¬ 
denser current is relatively quite small and for most practical 
purposes may be neglected. 

Having taken this glance at the inherent characteristics of 
condensers, we are prepared to investigate the effect of con¬ 
necting such condensers to the operating circuits of an asyn¬ 
chronous generator. 

Excitation Characteristics of Asynchronous Generators. 

Lines OL, OM and ON of Fig. 42 show the relation existing 
between the e.m.f. impressed upon certain condensers L, M 
and N and the current taken by them at constant frequency. 
As will be noted from the equations developed above, the effect 
of each condenser circuit can be represented by a right line 
passing through the origin and the slope of the line depends 
directly upon the am ount of e ffective condensance in the circuit 
considered. The line OPRTV is the excitation characteristic 
of an asynchronous generator when driven at constant speed 
without load and is found as the relation of the impressed e.m.f. 








INDUCTION MOTORS AS GENERATORS. 


87 


to the wattless, lagging, component of the current in the primary 
coil. 

If condensance of some value, such as M, be connected to the 
supply system in parallel with the asynchronous generator, then, 
at any impressed e.m.f., the condensance will require a definite 
amount of leading current while the lagging current of the 



Fig. 42. —Characteristics of Condensers and Synchronous 
Excitation Characteristics of Three-phase Machine. 


asynchronous generator will likewise be of a definite amount. 
With the impressed e.m.f. properly adjusted, the leading current 
taken by the condensance will equal the lagging current of 
the generator and no wattless current will be supplied from 
any other source. Such value of impressed e.m.f. is found in 
Fig. 42 at the intersection of the condenser line 0 M with the 
generator excitation characteristics at T, the value here being 




























































































































88 


ALTERNATING CURRENT MOTORS. 


approximately 292 volts. With conditions existing as here 
assumed, the asynchronous generator and condenser exciter 
may be isolated from the supply system and the set will auto¬ 
matically maintain its e.m.f. at 292 volts. 

If at the moment of disconnecting the asynchronous generator 
and the condensance M, as a unit, from the supply system, the 
impressed e.m.f. be 220 volts,—the normal value for the asyn¬ 
chronous machine,—the e.m.f. of the set will at once increase 
to 292 volts, due to the following causes: Under 220 volts the 
condensance takes 42 amperes leading current, which, when 
supplied from the asynchronous machine would raise its e.m.f. 
to 256 volts, at which e.m.f. the condensance would take 49 
amperes, raising the e.m.f. of the generator to 276 volts, and so 
on, until the e.m.f. became stable at 292 volts, as shown at T in 
Fig. 42. If, now, the condensance in circuit be changed to some 
value such as is represented by the line ON. the e.m.f. will at 
once rise to the value shown at the intersecting point, V. on 
the excitation characteristics. With a value of condensance 
such as OL in circuit, the generator will be unable to maintain 
its excitation at any e.m.f. whatsoever, since there is no point 
of intersection with the excitation characteristic. 

Due to the extremely weak magnetic condition in which a 
ring core without projecting poles is left when the exciting 
force is removed, and also to a large extent to the lower initial 
inverted knee of the excitation characteristic which requires a 
relatively large exciting force in comparison with the e.m.f. 
produced at this point, the generator possesses but slight ten¬ 
dency to build up from its remnant magnetism when con¬ 
densance of normal operating value is connected in circuit. 
It will be recalled that such behavior is characteristic also of 
shunt-wound, direct-current generators. In fact, for a given 
resistance in the shunt coil circuit, the relation between the 
current flowing therein and the e.m.f. is a right line similar to 
the condensance lines in Fig. 42 and the slope of such line deter¬ 
mine the e.m.f. up to which the machine will build. That the 
shunt circuit resistance must ordinarily be decreased below the 
operating value before the direct-current generator will build 
up from its residual magnetism is familiar to all. This is due 
to the fact that the slope of the shunt circuit current line is 
such as to cause it to intersect with the excitation characteristic 




INDUCTION MOTORS AS GENERATORS. 


89 


at the lower inverted knee of the initial line of the hysteresis loop. 

With the asynchronous generator, current of any frequency 
and of almost any value, or a static charge in the condensers 
may be used to cause the machine to build up. If the con¬ 
densance, when connected in the circuit, is above a certain 
value depending upon the magnetic circuit of the machine, the 
generator will build up instantly from its remnant magnetism 
without initial external excitation. 

Fig. 43 shows the connecting 'circuits for condenser excitation 
of an asynchronous generator and indicates a convenient method 
for varying the amount of effective condensance in circuit. 
For sake of simplicity, the diagram is made to represent single - 



Asynchronous 

Gen.exat.or 


p IG 43—Arrangement of Load and Exciter Circuits; Con¬ 
denser Excitation. 


phase circuits, although polyphase equipment throughout 
could similarly be employed. 

By the use of a transformer or auto transformer, the effective 
condensance in circuit can be adjusted within range, to any value 
desired by variation of the ratio of turns of the transformer 
coils, without in any manner changing the actual condensance 
connected thereto; the effective condensance varying as the 
square of the ratio of turns. This relation will be appreciated 
when it is considered that if, when the ratio is 1 to 1, the con¬ 
densance takes current /, when the ratio is increased to 1 to M, 
the primary e.m.f. remaining t he sa me as before, the secondary 
condensance current will be 1/ and the primary opposing 
current will b e~M times as great or M 2 1. It is thus seen that, 
although the source of excitation of the generator is the com 





























90 


ALTERNATING CURRENT MOTORS. 


densers, an increase in the step down ratio of transformation of 
the e.m.f. from the condensers will result in an increase in the 
generated e.m.f. 

Load Characteristics of Asynchronous Generators. 

The load characteristics of an asynchronous generator, when 
excited by static condensers, are quite similar to those of the 
familiar shunt-wound, direct current machine; an increase in 
load producing a decrease in e.m.f., due both to the direct effect 
of the load on the armature of the machine and to the added effect 
of the decrease in exciting current from the lessened e.m.f. at 
the field circuit. With the asynchronous machine a further 
effect is produced by the change in circuit frequency with the 
load, when the rotor is driven at constant speed. Since the 
effective condensance for any given adjustment varies directly 
w T ith the frequency, and the exciting current to produce a 
certain e.m.f. for the asynchronous machine varies inversely 
therewith, all other conditions remaining the same, a mere 
change in frequency will alter the e.m.f. at which the set will 
operate. Since even a non-inductive resistance drop of e.m.f. 
in the generator windings under load current causes a slight 
decrease of the current in the exciter circuit under the lessened 
terminal e.m.f., the best regulation is obtained when the con¬ 
denser is connected across one set of windings and the load 
placed across an independent set, as shown in Fig. 44. 

With the familiar synchronous alternator a lagging load 
current tends to decrease the terminal e.m.f. while a leading 
current has the opposite effect. An exactly similar state of 
affairs exists with a self-excited asynchronous generator. In 
this case, however, the effect is cumulative, since any change 
in the terminal e.m.f. causes a variation of the current in the 
condenser exciter and the field magnetism must adjust itself 
to a correspondingly altered valve. 

If it were possible always to operate the set at constant 
frequency, then, by use of Figs. 39 and 42, one could de¬ 
termine at once the amount of condensance necessary to main¬ 
tain the e.m.f. constant for any load, and conversely the in¬ 
herent regulation of the set could be ascertained. Take, for 
example, the condition of operation represented in Fig. 39, 
at 105.5 per cent, speed. The machine is delivering 16.5 horse- 


INDUCTION MOTORS A5 GENERATORS. 


91 


power at an efficiency of 80 per cent., and requires 42 amperes 
when the e.m.f. is 220 volts. The condensance necessary to give 
42 amperes leading current at 220 volts is shown in Fig. 42 
by the line O M. If the load be removed from the set and 
the circuit frequency remain constant, that is, if the rotor 
speed be decreased to 100 per cent., the e.m.f. will increase to 
292 volts as indicated by point T in Fig. 42. If, however, the 
rotor speed remained at 105.5 per cent, or increased when the 
load was removed, the e.m.f. would reach a much higher value. 
It is thus seen that close regulation necessitates that the ma¬ 
chine be operated above the knee of the excitation character- 



Fig. 44.—Exciter Circuits for Asynchronous Generator. 

istic (Fig. 42) and that the slip under load be small, that is, 
that the secondary resistance be small, as has been mentioned 

previously. 

Since magnetic saturation of material subjected to flux alter¬ 
nating at high frequency means excessive iron loss, efficiency of 
performance dictates that the core of an asynchronous generator 
be so designed that the secondary member, which is subjected 
to the low frequency of reversal corresponding to the slip 
reaches the saturation point while the primary member is yet 
much below such condition. 

When it is remembered that the amount of current taken by 
a condenser in an alternating-current circuit under a certain 























92 


ALTERNATING CURRENT MOTORS. 


e.m.f., varies directly with the frequency, and that the number 
of magnetic lines of force to produce a given e.m.f. in the gen¬ 
erator coils varies inversely with the frequency, it will be 
appreciated that the e.m.f. which a certain condenser will give 
to an asynchronous generator will depend largely upon the fre¬ 
quency. The frequency is determined primarily by the speed 
of the rotor, but it decreases, even for a constant rotor speed, 
when the generator load increases; a fact which shows the 
importance of good speed regulation at the prime-mover. Any 
leading current taken by the load acts as additional condenser 
capacity to increase the generated e.m.f., while lagging current 
produces the opposite effect; in fact, a condenser-excited gen¬ 
erator of this type which operates satisfactorily under a non- 
inductive load, may be caused to lose its e.m.f. entirely by the 
addition of a relatively small proportion of inductive load. 

Commutator Excitation of Asynchronous Generators. 

Mr. Heyland has devised a method for causing the asyn¬ 
chronous generator to supply its own current for excitation. 
In the application of his method, current is taken from the 
main circuit of the generator and passed to the secondary 
conductors through a suitable commutator on the rotor. The 
action of the commutator in supplying the exciting current wall 
be appreciated by first considering two extreme conditions of 
operation. If direct current be introduced by way of slip rings 
into the secondary windings of an asynchronous generator 
while the rotor is driven at normal synchronous speed, it will 
be found that alternating current at normal frequency may 
be obtained from the primary windings and that the e.m.f. 
generated may be adjusted in value by a corresponding change 
in the direct current supplied. In fact, one readily appreciates 
that operating under the conditions here assumed the asyn¬ 
chronous machine is converted into a simple alternating-cur¬ 
rent generator and possesses all of the characteristics of this 
type of machine. 

If with the rotor stationary alternating current of normal 
frequency be supplied to the secondary windings, current at 
the same frequency may be derived from the primary windings. 
Under these conditions, the generator is again a source of alter¬ 
nating current, but it now possesses the inherent characteristics 


INDUCTION MOTORS AS GENERATORS. 


93 


of a stationary transformer. If now a commutator be con¬ 
nected to the secondary windings, any motion of the rotor in 
either direction will not alter the effect of any certain value of 
current in the secondary in producing e.m.f. in the primary 
winding, since the space-phase position of the current with refer¬ 
ence to the primary coils will be the same as would be the case 
were the rotor stationary. Hence, independent of the speed 
of the rotor, the current thus introduced into the secondary 
reacts upon the primary with the primary frequency. The 
value of the current in the secondary can be varied by changing 
the e.m.f. impressed upon the commutator, and it may be given 
any phase position with reference to its reaction upon the 



Compound Excitation 

Fig. 45.—Asynchronous Generator; Commutator Excitation. 

primary by changing the position of the rotor brushes relative 

to the field coils. (See Fig. 45.) 

When the rotor is driven at synchronous speed in the direc¬ 
tion of the revolving field, the e.m.f. required to be impressed 
upon the secondary windings in order to cause a given current 
to flow there through is greatly reduced below the value neces¬ 
sary when the rotor is stationary, due to the practical elimina 
tion of the reactive e.m.f. at the low frequency of the current 
in the individual slots containing the secondary conductors. 

The operation of an asynchronous generator with commutator 
excitation is quite similar to that of one excited by means of 
condensers, and the statements made above as to the effect 
of speed variations and of the character of the load current 

















94 


ALTERNATING CURRENT MOTORS. 


upon the delivered e.m.f., and as to the necessity of working 
the magnetic core at high density apply equally to the Heyland 
shunt-excited asynchronous generator. With a commutator 
generator, however, it is possible to use compound excitation, 
which consists in passing the load current, by means of an 
auxiliary set of brushes on the commutator, through the sec¬ 
ondary conductors, and thus to increase the field strength 
with the load and to counteract the effect of any lagging current 
upon the field magnetism and the circuit e.m.f. This latter 
action is very similar to that produced in direct-current gen¬ 
erators equipped with the Ryan balancing coils. Since a load 
current which lags in the primary coils will lag equally in the 
auxiliary exciting circuit of the secondary, it is possible by 
means of the compound excitation to compensate for the field 
demagnetizing effect of an inductive load upon the generator. 


CHAPTER VIII. 

TRANSFORMER FEATURES OF THE INDUCTION MOTOR. 


Electric and Magnetic Circuits. 

Fig. 46 represents the electric and magnetic circuits of an ideal 
transformer. When an alternating e.m.f. is impressed upon the 
primary terminals, the secondary being on open circuit, a certain 
value of current exists in the coil. The magnetomotive force 
due to the ampere-turns of this current causes lines of force to be 
produced in the core, and the change in the value of these lines with 
the alternation of the current generates in the primary coil an 
e.m.f. in a time-phase position to tend to decrease the current 
in the coil; the final result being that there flows in the coil, 



Fig. 46.—Electric and Magnetic Circuits of an Ideal Transformer. 

just that value of current whose product with the number of 
primary turns gives the magnetomotive force necessary to send 
through the reluctance of their path that number of lines 
the change in the value of which generates in the primary coil 
an e.m.f. less than the impressed by an amount just sufficient 
to allow this value of current to flow through the local im¬ 
pedance of the primary coil. If the local impedance of the 
primary coil—which is composed of the resistance and the 
local reactance due to the leakage lines surrounding only this 
coil—be of small value, the e.m.f. counter generated in the 

95 
















































96 


ALTERNATING CURRENT MOTORS. 


coil by the alternating flux in the core will be practically equal 
to the impressed e.m.f. 

The effect of varying the reluctance of the magnetic path in 
the core is to vary accordingly the exciting magnetomotive 
force, but no appreciable effect is produced upon the value of 
the flux. A negligible effect may be attributed to the changed 
value of exciting current through the slightly varied local re¬ 
actance and the resistance of the primary coil, which may 
alter the diminutive loss of e.m.f. through this impedance. 
This effect, which throughout the operating range of well de¬ 
signed transformers is augmented to a negligible extent by 
the load current, will be treated more in detail later. 

The secondary coil is placed mechanically in a position to be 
cut by the greatest proportion of the flux due to the primary 
exciting magnetomotive force. With this coil on open circuit, 
there will be generated in each of its turns by the change in the 
va’ue of core flux, an e.m.f. equal to the counter e.m.f. per turn in 
the primary. If the secondary circuit be closed through an im¬ 
pedance, current will flow, due to the secondary e.m.f., and this 
current will tend to decrease the core flux. The e.m.f. counter 
generated in the primary being somewhat lessened, more cur¬ 
rent will flow therein tending to restore the flux to its former 
value, and stable conditions w T ill be reached when the additional 
primary current has a value and phase position such as to give 
the magnetomotive force necessary to counterbalance the effect 
of the secondary ampere turns, thus keeping the flux in the core 
quite closely constant at the value demanded by the primary 
e.m.f. 

The exact relation between the flux in the core, the frequency 
of the supply current, the primary counter e.m.f. and number 
of turns can be derived quite simply from that law of physics 
which states that one c.g.s. unit of e.m.f. is generated when 
flux cuts a conductor at the rate of one line per second. In 
general 



Assuming the flux (and the e.m.f.) to vary with time in a 
manner to be represented by the familiar sine law, its value can 
be stated as 

<t> = A B m sin co t 



TRANSFORMER FEATURES OF INDUCTION MOTOR. 97 


where A B m is the maximum value of the total flux over the area 
of the core A. 

Thus there is obtained 

d ( A B m sin co t) 

@ - 
d t 

e — A B m co cos co t 

which becomes maximum when cos co t = 1 or 

Cfyi A. B yyl CO 

so that the virtual value of the e.m.f. per turn expressed in 
c.g.s. units is 

A. B yyi CO 

/— 

v 2 

which for N turns when expressed in volts becomes, 

A B m 2 r, j N A B m f N 

\/2 10 8 10 8 

from which the total magnetic flux, A B m , or the flux density, 
B m , may be determined. 

Due to the internal friction in turning the molecular magnets 
in first one and then the other direction with the alternation of 
the core flux, there is dissipated a certain amount of energy 
with each reversal of flux, such energy appearing as heat in the 
core material and requiring in the primary coil a certain com¬ 
ponent of current in phase with the impressed e.m.f. to supply 
this loss. The watts thus required vary with the 1.6 power 
of the flux density and with the quality of the magnetic mate¬ 
rial, and can be expressed thus, as shown by Dr. C. P. Steinmetz, 

z } V B J* 

Wh 10 7 

where B m = maximum flux in lines per sq. c.m. 

V = volume of core in cubic c.m. . 

/ = frequency in cycles per second. 
z = coefficient of hysteresis. 

z varies from .001 to .006 according to the quality of the 
magnetic material, a fair value for transformer sheets being .0022. 







98 


ALTERNATING CURRENT MOTORS. 


The alternation of the flux in the thin sheets (14 mils) com¬ 
prising the transformer causes the generation within each sheet 
of a minute value of e.m.f. which tends to send current through 
the conducting material of the sheet. This current in its pas¬ 
sage through the sheet follows the laws common to all electric 
circuits and produces heat proportional to the square of its 
value and the resistance through which it passes. The watts 
thus dissipated may be expressed as 

_ e (d f B m ) 2 V 
10 11 


where d = thickness of sheets in centimeters, 
e = coefficient of eddy loss. 

e varies with the specific conductivity of the core material, a 
fair value being 1.65. 

The eddy current and hysteresis losses being similar in effect 
are frequently treated as one quantity under the term core 
losses, requiring in the primary coil a current in phase with the 
impressed e.m.f., and of a value such that its product with the 
e.m.f. gives the core loss watts. 

While the value of the flux in the core is determined almost 
exclusively by the primary e.m.f. the number of turns and the 
frequency, quite independent of the permeability of the magnetic 
path, the value of the magnetomotive force to produce such flux 
is directly dependent upon the permeability and varies inversely 
therewith. A convenient method for obtaining the value of the 
magnetomotive force expressed in ampere turns is found from 


the fact that one ampere turn produces 



% 

lines per centimeter 


cube of air. From this fact it follows that one ampere turn pro- 
1 25 u A 

duces —~—j -— ^ nes a material of permeability /*, whose length 


of magnetic path is l centimeters and cross sectional area A sq. 
centimeters. A convenient method for determining the exciting 
watts from the volume of the core will be discussed later. 


Equivalent Electric Circuits. 

For the magnetic and electric circuits of a transformer as 
represented in Fig. 46 may be substituted the equivalent electric 
circuits shown in Fig. 47; 'where R p and X P are the primary re- 





TRANSFORMER FEATURES OF INDUCTION MOTOR. 99 


sistance and local leakage reactance, while R s and X s are the 
secondary resistance and local leakage reactance, the shunted 
inductive and non-inductive circuits carrying the exciting cur¬ 
rent and core loss current, respectively. These are the true 
equivalent circuits of a transformer based upon a ratio of pri¬ 
mary to secondary turns of 1 to 1. If the primary has n times 
as many turns as the secondary, then the same equivalent 
circuits may be used to represent the transformer, if the actual 
secondary resistance and local leakage reactance be multiplied 
by n 2 to obtain the values to be used in the equivalent circuits 
and the real secondary load current be divided by n. 

In the non-inductive shunt circuit flows the current to supply 
the core losses. If these losses varied as the square of the in¬ 
ternal counter e.m.f., that is, as the square of the magnetic 
flux, the circuit could be considered as composed of true re- 



Di 

Fig. 47.—Equivalent 


D D 2 

Electric Circuits of an Ideal Transformer. 


sistance. Such, however, is true only with reference to the 
eddy current loss and is not directly applicable to the hysteresis 
loss, but the assumption of the existence in this circuit at all 
times of a current whose product with the voltage supplies the 
core losses eliminates any error due to treating the circuit as 
being of pure resistance. 

Modified Electric Circuits. 

It is obviously possible to derive readily complete equa¬ 
tions representing the performance of the transformer under 
various conditions of load by the use of the circuits shown in 
Fig. 47, when proper values are assigned to the several constants 
there indicated. There may, however, be introduced in the 
arrangement of the circuits a slight modification which in¬ 
volves no measurable error and yet which allows the performance. 




















100 


ALTERNATING CURRENT MOTORS 


of the transformer to be represented graphically by a diagram 
whose most prominent feature is its simplicity. In Fig. 48, 
the two shunted circuits are shown as connected in the supply 
line so as always to receive the full value of the impressed 
e.m.f. The magnitude of the error thus produced will be ap¬ 
preciated when it is recalled that the current for supplying the 
core losses and the exciting current taken by a transformer 
are in any case quite small and the assumption that the com¬ 
bined value of such small currents remain constant when in 
reality it varies inappreciably (seldom over two per cent.) 
with the load leads to a truly negligible discrepancy in the 
results thus obtained. 

With connections made as indicated in Fig. 48, the current 
taken by each of the three circuits will flow independently of 



Fig. 48. —Practically Exact Representation of Circuits of a 
Transformer or Induction Motor. 


the currents in the other two circuits while the total measurable 
primary current will be the vector sum of the three components. 
Methods for determining the value of the current for supplying 
the core losses and the exciting current have been given. 

Circle Diagram of Currents. 

The current taken by the load flows through the local im¬ 
pedance of both the primary and secondary coils and is unaffected 
by the presence of the currents in the other shunted circuits. 
If the external load circuit be strictly non-inductive, the locus 
of the load current with change in the resistance will be the arc 
of a circle, whose diameter is the ratio of the primary e.m.f. 
to the sum of the local reactances of the primary and secondary 
coils. (1 to 1 ratio.) 






















TRANSFORMER FEATURES OF INDUCTION MOTOR. 101 


Let R l be any chosen value of load resistance, then the load 
current will be 

j __ Ep _ 

s/ (^l+ Rp-u R s y + (X P +x s y 


and its phase position with reference to the primary e.m.f. E P 
will be such that 


hence 


sin 0 


X P AX S 

V (R l + Rp-\-R s ) 2 + {X P + X s ) 2 



which when E P , X P and X s are constants is the polar equation 
of a circle having diameter 

E p 

X p + X s 


Knowing the values of the three component currents in the 
branch circuits of Fig. 48, the resultant primary current may be 
found in value and phase position as their vector sum. Since 
the exciting current and the core loss current do not change 
with variation in the load, their vector sum may be perma¬ 
nently recorded as the no-load primary current as shown at 
M 0 in Fig. 49. It is convenient also to plot at once the vector 
of the load current 0 P in position to give at once the resultant 
primary current, by beginning the current locus 0 P C at the 
point 0 in Fig. 49. 

A study of the construction of the current locus of Fig. 49 
will show that at any chosen load current, as 0 P, M P is the 
resultant primary current, G M P is the angle of lag of the 
primary current behind the e.m.f. M E , and the ratio of M G 
to M P is the power factor. The product of M G and the im¬ 
pressed primary e.m.f. is the input to the transformer, from 
which if the core losses and the primary and secondary copper 
losses be subtracted the output may be obtained and the effi¬ 
ciency determined. The output divided by the secondary 
current givos the secondary e.m.f. from which the regulation 
of the transformer may be ascertained. 









102 


ALTERNATING CURRENT MOTORS . 


In a well constructed static transformer, the primary and 
secondary coils are so interspaced that magnetic leakage is 
reduced to a minimum*, so that X P and X s are small in value 
and the diameter of the current locus as shown in Fig. 49 is 
correspondingly enormous, and throughout the operating range 
of the transformer the arc of the circle deviates but slightly 
from a straight line parallel to M E. Numerous theoretical 
equations are available for determining the values of X P and X s . 
Only those which are formed upon an experimental basis in¬ 



volving the use of the exact type of transformer under con¬ 
sideration are found to give results in conformity to observations 
under test. 

On account of the fact that the path of the magnetic lines in 
the core of a static transformer pass through material of high 
permeability a relatively small value of exciting magnetomotive 
force is required,and due also to the low value of the core losses 
the no load current of such a transformer is correspondingly 
small in comparison with the full load current. A type of 






TRANSFORMER FEATURES OF INDUCTION MOTOR. 103 


transformer in which the proportions of no load current to full 
load current is quite large is found in the induction motor. In 
its electrical characteristics an induction motor is essentially a 
transformer possessing high magnetic leakage due to the separa¬ 
tion of the primary and secondary windings and the more or 
less complete surrounding of each group of coils with material 
of high permeability. This transformer shows also a high value 
of exciting current due to the double air gap in the magnetic 
circuit. For studying the characteristics of such a transformer 
the circle diagram is especially valuable. 

Although in the secondary circuit the frequency varies 
directly with the slip, and, therefore, for constant coefficient of 
self-induction, the secondary reactance varies in direct propor¬ 
tion with the slip, and in general the secondary resistance is 
more or less constant, it is convenient and helpful to treat the 
machine as a stationarv transformer in which the secondarv 
reactance is constant and the resistance varies with the load. 
That is to say, the performance of the secondary of the motor 
will be faithfully represented if all the power received from the 
primary be considered as dissipated in resistance in the sec¬ 
ondary circuit, the slip at all times being taken as unity, and 
the total secondary resistance (conductance) being assumed to 
be varied according to the secondary load, or, briefly, the in¬ 
duction motor may be treated in all respects like a stationary 
transformer. The effect upon the transformer quantities of 
increasing the speed from zero to synchronism is the same in 
all respects as increasing the external resistance in the sec¬ 
ondary circuit from zero to infinite value, the output from the 
motor being represented in any case as the power lost in the 
fictitious external resistance. These facts have been discussed in 
a preceding chapter and they need not be further discussed here. 

The diagram of Fig. 47, which shows the conventional method 
of representing the electric circuits of a transformer which has 
an equal number of turns in the primary and secondary wind¬ 
ings, is based on the assumptions that the reluctance of the 
core is constant for all densities of magnetisms and that the 
iron losses vary with the square of the magnetic density in the 
core. It is obvious that neither of these assumptions is abso- 
lutelv correct with reference to any commercial stationary 
transformer, but it is true that the errors involved in any 


104 


ALTERNATING CURRENT MOTORS. 


calculations depending upon these assumptions are practically 
negligible. It is evident that in an induction motor the re¬ 
luctance of the total magnetic path is much more nearly con¬ 
stant than that in a transformer, and that if the frictional 
and windage losses be included with the iron losses, the circuits 
shown in Fig. 47 will serve admirably for all calculations con¬ 
nected with this machine. 

In the diagram of Fig. 47, R P and X p are the primary re¬ 
sistance and local leakage reactance, while R s and X s are the 
secondary resistance and local leakage reactance, the shunted 
inductive and non-inductive circuits carrying the exciting cur¬ 
rent and the core loss current, respectively, as stated previously. 
An examination of Fig. 47 wiil show that the primary current 
is made up of three components; the quadrature exciting cur¬ 
rent, the core loss current and the load current, of which it is 
the vector sum. Under operating conditions the current which 
flows through the primary coil causes a drop in voltage across 
the local primary impedance and hence the internal counter 
e.m.f. decreases with increase of load, and there is a decrease 
in both the exciting current and the core loss current. If it be 
assumed initially that the variation in the values of these cur¬ 
rents is negligible in comparison to the load current of the 
machine, the treatment becomes much simplified and yet the 
true conditions are fairly well represented. 

Fig. 48 shows the circuits as they could be represented on the 
basis of the latter assumptions. The current taken by each of 
the three circuits will flow independently of the currents in the 
other two circuits, while the total measurable primary current 
will be the vector sum of the three components. Only one of 
the component currents varies with the change in load, and its 
value can easily be determined when the resistance of the load 
circuit is known. It will be noted that the load circuit con- 
ains a constant reactance (Xp H- Xs) in series with a variable 
resistance (Rp + Rs-\-R L ), where R L is the fictitious resistance 
of the load. It will be seen, therefore, that the current which 
flows through this circuit under a constant impressed e.m.f., 
E , can be represented by a vector whose extremity describes 
the arc of a circle having a diameter equal to 

E 


Xp + Xs 



TRANSFORMER FEATURES OF INDUCTION MOTOR. 103 


The arc 0 P C of Fig. 49 indicates a section of such a sec¬ 
ondary current locus. At any point, P , on this arc, O P repre¬ 
sents the secondary current both in value and phase position. 
The quadrature exciting current is represented by 0 N, while 
N M shows the core loss current (to supply all of the no-load 
losses). The vector sum of these three currents, M P , in Fig 
49, is the primary current, while the angle of lag of the current 
behind the circuit e.m.f. is shown by N M P, the cosine of 
which is the power factor. The power component of the pri¬ 
mary current is indicated by the line P Q, and the product of 
this with the circuit e.m.f. gives the input to the motor. 

By means of this simple circle diagram, the construction of 
which is based on somewhat erroneous assumptions, the com¬ 
plete performance of the motor may be determined with a 
degree of accuracy which seldom need be exceeded for any pur¬ 
pose of designing or testing, since the errors introduced are of 
small moment, and are not misleading; moreover, they tend to 
disappear in the final composite results. Thus the input to 
the secondary at any chosen current may be found as the 
difference between the primary input and the sum of the pri¬ 
mary losses, which latter include the easily calculated copper 
loss and the approximated “constant ’’ losses. The secondary 
input is at once the torque in “ synchronous watts the ratio 
of the secondary copper loss to the secondary input is the slip, 
while the output is the secondary input minus the secondary 
copper loss. As will be shown later, the various quantities 
may be represented graphically by simple circular arcs and 
straight lines. 

Internal Voltage Diagram of the Induction Motor. 

In comparing Fig. 48 on which the simple circle diagram of 
Fig. 49 is based, with Fig. 47 on which a true diagram should 
be based, it will be observed that the errors involved relate merely 
to the quadrature exciting current circuit and the core loss 
current circuit; the voltage across these circuits is not con¬ 
stant, but it varies with the load current. Since that portion of 
the drop of voltage across the primary impedance which is 
due solely to the core loss current and the exciting current, is 
quite negligible in comparison to that due to the local current, 
it is permissible to assume that the voltage at B D in Fig. 47 


106 


ALTERNATING CURRENT MOTORS. 


depends entirely upon the load current. This assumption is 
equivalent to neglecting terms of higher order. These may be 
taken into consideration graphically without difficulty, but the 
gain by so doing is not sufficient to justify the added com- 
plications. 

The internal voltage diagram of an induction motor is repre¬ 
sented in Fig. 50, where A D is the impressed primary e.m.f., 
A F is the drop through the primary reactance, B F is the drop 
through the primary resistance, and, hence, A B is the primary 
impedance drop. The secondary reactance drop is B H, C H 



Figs. 50 and 51.—Modified e.m.f. and Current Loci of Poly¬ 
phase Induction Motor. 


is the secondary resistance drop, and B C is the secondary im¬ 
pedance drop. CD is the voltage consumed in the fictitious 
external load resistance, and, hence, is the secondary e.m.f.; 
Es in Fig. 48 or Fig. 47. The line B D in Fig. 50 is the e.m.f., 
E e in Fig. 48 or the voltage across B D in Fig. 47. 

The current in the secondary is in phase with G D, and in 
quadrature with A G. The angle, A G D, is a right angle, so 
that the point, G, describes a circle with its center on the line, 
A D. The point, F, describes a circle, A F I, with its center 
at point, /, on line, AID. The distance, A I, bears to the 













TRANSFORMER FEATURES OF INDUCTION MOTOR. 107 


distance, A D, the ratio of the primary leakage reactance to the 
total leakage reactance of the machine. The point, B, describes 
a circle, A B I L. The angle, LAI, has a tangent equal to 
the ratio of the primary resistance to the primary leakage 
reactance. 

It will be noted from Fig. 47 that the three component currents 
may be determined at once when the secondary load current is 
known and the voltage across B D has been found. Referring 
now to Fig. 52 and remembering that the core loss current is in 
phase with B D and proportional to it, it will be seen that this 
current can be represented by the line, D T, where T describes 
a circle having its center on the line, V X, the angle, D V X, 
being equal to the angle 1 A L. Similarly, the exciting cur¬ 
rent, which is in quadrature with B D, can be represented by 
the line, D S, where S describes a circle having its center on the 
line, U W, the angle, D U W, being equal to the angle, I A L. 
D U and D V, of Fig. 52, are respectively equal to D U and 
D V of Fig. 50 or to 0 N and N M of Fig. 51. 

Corrected Current Locus of the Induction Motor. 

The corrected current locus of the induction motor is shown 
in Fig. 53, where the arc, 0 P R, is in all respects the same as the 
semicircle, OPR, in Fig. 51. The line O P in Fig. 53 is parallel 
to the line, D G, in Fig. 52, but it varies in length directly with 
the line, A G, of Fig. 52. N M of Fig. 53 is equal to D T, and 
is parallel and proportional to B D of Fig. 52. 0 N is equal to 

D S, is in quadrature to and proportional to B D of Fig. 52. 
The primary current is represented by the line, M P, as the 
vector sum of 0 P, 0 N and N M, that is, as the resultant of 
the load current, the quadrature exciting current and the core 
loss current. 

In Fig. 53, the line P Q is the power component of the pri¬ 
mary current, the product of which with the impressed e.m.f. 
gives the input to the motor. The complete performance of 
the machine can be determined in a manner exactly similar to 
that outlined for the simple circular locus of Fig. 51 or 49. 

The current locus in Fig. o3 is based primarily on the trans¬ 
former features of the induction motor, and it is inexact only 
to the extent to which the motor differs from a transformer in 
its electrical behavior. The frictional loss is treated as a core 


108 


ALTERNATING CURRENT MOTORS. 


loss, and hence it is tacitly assumed that this loss necessitates 
a power component of current in only the primary winding. 
This treatment involves no error with reference to the primary 
current, but it neglects a certain component of secondary cur¬ 
rent, which, however, is too small to need consideration. 

It will be noted from Fig. 50 that the secondary resistance 
has no effect whatsoever on either the current locus shown in 
Fig. 51, or that shown in Fig. 53. The primary resistance has 
no effect on the secondary current, although the primary 


A 



Figs. 52 and 53. —Corrected e.m.f. and Current Loci of Poly¬ 
phase Induction Motor. 

current depends somewhat on this resistance. If the primary 
resistance were negligible, the point B in Fig. 52 would follow 
the circular arc, A F 7, and both the angle, D U W , and the 
angle, D V X, would reduce to zero; the general form of the 
locus of Fig. 53 would be changed only slightly. 

From the facts stated above, it is evident that the primary 
current cannot be represented by any circle howsoever located, 
but that in any event the secondary current locus is a true circle 
For many purposes where extreme accuracy is not desired, but 
where some information is wished concerning the changes in 







TRANSFORMER FEATURES OF INDUCTION MOTOR. 109 


the variables connected with the phenomena of an induction 
motor during operation an approximate graphical diagram 
without serious errors is extremely convenient. It is believed 
that the simple circular locus with its well defined but practi¬ 
cally negligible errors possesses peculiar merit in this respect. 

Complete Performance Diagram of the Polyphase Induc¬ 
tion Motor. 

A simple circular locus from which the complete performance 
of a polyphase induction motor may be ascertained at once is 
shown in Fig. 54. At any point P on this locus, the line M P 
represents the primary current, while the angle, E M P is the 
angle of lag of the current behind the primary e.m.f., E M . 
The line 0 P shows the secondary current, both in value and 



Fig. 54. —Simple Circular Current Locus of a Polyphase 

Induction Motor. 

phase position. When the secondary current has zero value, 
that is, at synchronous speed, the primary current becomes 
equal to M O, 0 N being its “ wattless ” component and N M 
its power component to supply all of the no-load losses. When 
the rotor is stationary, the secondary current assumes some 
value such as O F, and the primary current is the vector sum 
of O F and OM (not drawn). The curve, 0 P F, is the arc 
of a circle having its center on the line 0 N prolonged. 

Under starting conditions, all of the power received by the 
motor is used in supplying the copper and core losses of the 
machine. The line F I is the power component of the primary 
current at starting, and hence, by the use of the proper scale, 
it may represent the total losses of the machine when the rotor 
















no 


ALTERNATING CURRENT MOTORS. 


is stationary. If it be assumed that the circuits of the machine 
are faithfully represented by Fig. 48, then the line H F = 
{F I — M N) of Fig. 54, shows the secondary copper loss and 
the increase of the primary copper loss over its (synchronous) 
no-load value. The line H F being properly divided at G, G H 
represents the increase of the primary copper loss, and G F 
the secondary copper loss for the current 0 F. By drawing 
from the point 0 a straight line, 0 G J, passing through the point 
G, the complete performance of the machine may be determined 
directly from inspection. 

If from any point P on the circular locus a line be drawn per¬ 
pendicular to the diameter, 0 K, the following quantities may 
be observed at once: 

M P is the primary current, 

E M P is the primary angle of lag, 
cos E M P is the power factor, 

0 P is the secondary current, 

P T is the total primary input, 

T S is the “ constant ” losses of the machine, 

R S is the added primary copper loss, 

R T is the total primary losses, 

PR is the total secondary input, in watts, 

P R is the torque in synchronous watts, 

Q R is the secondary copper loss, 

Q R-t- PR is the slip, with synchronism as unity, 

Q P-r-P R is the speed, 

Q P is the output, 

Q P + P E is the efficiency. 

The maximum power factor occurs when the point P is at A„ 
where the line M P becomes tangent to the circle. 

The maximum output occurs when P is at B , the point of 
tangency of a line drawn parallel to 0 F. 

The maximum torque occurs when P is at C, the point of 
tangency of a line drawn parallel to 0 G. The current which 
is required to give maximum torque varies somewhat with the 
primary resistance, but it is independent of the secondary re¬ 
sistance, although the speed at which’ the maximum torque is 
obtained depends largely on the value of the secondary resist¬ 
ance. The secondary resistance required to give maximum 
torque at starting bears to the assumed constant primary re¬ 
sistance the ratio of G' C to G f FL r of Fig. 54. 


TRANSFORMER FEATURES OF INDUCTION MOTOR. Ill 


The proof of the accuracy of the diagram in representing the 
value and phase positions of the primary and secondary currents, 
for the circuits as shown in Fig. 4S was given above, and it 
need not here be repeated. That the various other quantities 
are accurately represented as indicated may be shown as follows: 

Denoting as a the angle P 0 K of Fig. 54, it will be seen that 
0 S = 0 P cos a , and that 0 P =0 K cos a. Hence, 0 S = 

0 K cos 2 a, or OS = O P 2 + 0 K. The interpretation of the last 
equation is that, as the point P moves around the circle, the 
line 0 S is at all times proportional to the square of the line 0 P. 
That is to say, the line O S is proportional to the secondary 
copper loss or to the increase in the primary copper loss over 
its no-load value. Under starting conditions, the secondary and 
the added primary copper losses are represented by F G and G H ; 
and at any point P, the corresponding losses must bear to F G 
and to G H the ratio of 0 S to 0 H. Therefore, the secondary 
and the added primary copper losses are accurately shown by 
the lines Q R and R 5, respectively. 

That the ratio of the secondary copper loss to the total secondary 
input is equal to the slip has already frequently been metioned. 
It seems desirable, however, in this connection to call attention 
to the fact that this ratio is a true measure of the slip whether 
the magnetism of the machine remains constant or not, and 
that the accuracy of the determination of the slip by this method 
is in no wise affected by the substitution of the modified circuits 
of Fig. 48 for the exact circuits of Fig. 47. Thus, if the secondary 
copper loss is determined without error, and the secondary input 
is known, both the speed and the torque may be ascertained with 
precision. It is seen, therefore, that any errors introduced 
must relate to either the currents or the losses. 

If the points 0 and F of Fig. 54 are obtained from an actual 
test on a machine, it is evident that the circle diagram as con¬ 
structed must be at least approximately correct for the primary 
current locus. Under starting conditions the power received 
by the machine is accurately represented by the line FI. If 
the distance F G be drawn equal to the easily determined 
“ added ” primary copper loss, the distance G H must represent 
the secondary copper loss with a fair degree of accuracy. 

It is especially worthy of note that under starting conditions 
and at synchronous speed the errors are eliminated, and through- 



112 


ALTERNATING CURRENT MOTORS 


out the operating range of the motor the various errors tend to 
, cancel each other. Even in extreme cases, where the (syn¬ 
chronous) no-load triangle, 0 M N, is large in comparison with 
the circle diagram, the errors are relatively small and for most 
practical purposes may well be neglected. 

It is not possible to obtain absolute accuracy in a simple 
diagram of an induction motor. Moreover, it is unnecessary to 
construct a diagram with a degree of accuracy greater than that 
which can be employed with it in scaling off the various values. 
The principal advantage to be found in the graphical method of 
treating induction motor phenomena resides in the fact that by 
the use of a simple diagram one is able to follow optically, and 
thus mentally, the changes which take place throughout the 
operation of the machine, while in the manipulation of algebraic 
formulas, which can be used for absolute accuracy, one is apt 
to find himself more or less in the dark concerning these changes. 

The above description refers to the locus of the primary and 
secondary currents of a polyphase induction motor. It is de¬ 
sirable to describe also the method by which a similar diagram 
may be used with a single-phase induction motor. 

Comparison of Single-Phase and Polyphase Motors. 

The chief difference between a single-phase and a polyphase 
induction motor resides in the character of the magnetic fields 
of the two machines. At synchronous speed each machine 
possesses a true revolving field. At standstill, however, while 
the magnetic field of the polyphase motor revolves synchron¬ 
ously and is of more or less constant strength, the field of the 
single-phase machine is unidirectional in space and alternating 
in value. See Chapter V. 

If when the rotor of a polyphase motor is stationary a circuit 
be opened so that current flows through only two leads of the 
machine, it will be found that the total volt-amperes taken by 
the machine decrease to about one-half of the former value, 
the power factor being practically unchanged. That the mag¬ 
netomotive force of the current in each phase winding of a two- 
phase motor when the rotor is stationary produces a flux which 
(for constant reluctance of the core) acts as though the flux 
due to the other current in the winding were not present may 
be seen at once without proof. It will be appreciated, also, 


TRANSFORMER FEATURES OF INDUCTION MOTOR. 113 


that the currents produced in the secondary by two alternating 
fluxes which are in electrical space quadrature do not interfere 
one with the other, so that the current in each primary winding 
flows just as though the other primary current did not exist. 
Thus the “ equivalent single-phase ” starting current of a two- 
phase motor is just twice that of the same motor when only 
one phase winding is used; the power factor is the same in the 
two cases. Both experimental and theoretical investigations 
show that the “ equivalent single-phase ” starting current of a 
three-phase motor is also equal to twice the current which flows 
through two leads when the third lead is interrupted. 

If, when the rotor of a polyphase induction motor is revolving 
synchronously a primary circuit of the machine be opened, it 
will be found that the current flowing through the remaining 
leads increases somewhat but that the total volt-amperes taken 
by the machine remain practically constant, and the power- 
factor is practically unaltered (the power component of the 
equivalent single-phase current increases to a small extent while 
the wattless component decreases slightly). The action of the 
machine at synchronous speed is attributable to the continued 
existence of a revolving magnetic field or practically constant 
strength which requires a definite component of current in 
phase with the voltage to supply the losses and another com¬ 
ponent in quadrature with the voltage to supply the “ quadrature 
watts ” for excitation. A subsequent chapter will explain the 
distribution of current in the secondary conductor, and will 
show in what manner the “ quadrature watts” for the “ speed- 
field” are supplied by the primary exciting magnetomotive 
force. 

When' the rotor of a polyphase motor is revolving synchron¬ 
ously, the secondary current has a negligible value. In the 
single-phase motor, however, the secondary current at synchron¬ 
ous speed has a value such that its magnetomotive force produces 
in electrical space quadrature with the main alternating field 
through the primary coil, a field which is equal in value and 
in time quadrature with the main field. The value of the main 
field is determined by the primary e.m.f. just as is true in any 
transformer, while the field which is in quadrature both in time 
and in space therewith depends for its value both upon the 
“ transformer field ” and upon the speed of the rotor; the two 


114 


ALTERNATING CURRENT MOTORS. 


fields are equal in effective value at synchronous speed and at 
other speeds, the “ speed field ” is equal to the “ transformer ” 
field multiplied by the speed. Thus the “ speed-field ” com¬ 
ponent of the secondary current varies with the speed and is 
zero at standstill. 

Electric Circuits of the Single-Phase Induction Motor. 

The circuits of a single-phase induction motor can be repre¬ 
sented with a fair degree of accuracy if the primary and sec¬ 
ondary resistances and the leakage reactances be arranged as 
shown in Fig. 55a, the “ transformer field ” and “ speed field ” 
exciting circuits being connected as indicated. The current 
taken by the load and that used to produce the “ speed field ” 
pass through both the primary and the secondary coils, while 
the current required for the “ transformer field ” flows through 



Dj d r 2 

Fig. 55a. —Practically Exact Representation of Circuits of 
a Single-phase Induction Motor. 

only the primary coil. When the load circuit is opened, that 
is, at synchronous speed, the “ speed field ” current and the 
“ transformer field ” current are practically equal in value. 
When the resistance of the load circuit is zero, that is at stand¬ 
still, the “ speed field current is zero, and the current which 
flows through the coils of the machine acts as though the “ speed 
field ” circuit were not present. 

It is to be noted especially that the decrease in the “ speed 
field ” current below the value of the “ transformer field ” cur¬ 
rent is attributable to the variation of the rotor speed from 
synchronism and not to the drop in voltage across the secondarv 
winding, which is caused by the load current. The “ secondary 
load ” current flows in electrical space quadrature to the “ speed 
field ” current, and the two currents in no way interfere with 
each other. The statements just made relate exclusively to 
the current whose magnetomotive force produces the “ speed 





















TRANSFORMER FEATURES OF INDUCTION MOTOR. 115 


field,” which current, on account of its electrical space position, 
does not react in any way upon the “ transformer field.” The 
secondary carries also another component of current in addition 
to the load current. The time-phase position and the electrical 
space positions of this component are such that its magneto- , 
motive force tends directly to decrease the “ transformer 
field thus, it acts like a “ wattless ” secondary current. It 
is this latter component of secondary current which is represented 
in the circuit diagram of Fig. 55a. This component bears to the 
actual “ speed field ” current (approximately) the ratio of the 
actual rotor speed to the synchronous speed. Thus the voltage 
impressed upon the “ speed field ” circuit of Fig. 55a is (approx¬ 
imately) equal to that impressed upon the “ transformer field ” 
circuit multiplied by the square of the speed, synchronism being 



Fig. 55b. —Modified Representation of Circuits of a Single¬ 
phase Induction Motor. 

taken as unity. These facts will be discussed more fully in a 
subsequent chapter. 

Although it is possible to construct primary and secondary 
current loci based on the circuits shown in Fig. 55a, the problem 
of dealing with the value and phase positions of the currents is 
greatly simplified without involving a detrimental loss of accu¬ 
racy by using the modified arrangement of circuits indicated 
in Fig. 55b. 

Complete Performance Diagram of the Single-Phase In¬ 
duction Motor. 

The current diagram for the circuits shown in Fig. 55b is 
given in Fig. 56, where M N is the power and 0 N the wattless 
component of the primary current at synchronous no load, while 
FI is the power and I M the wattless component of the pri¬ 
mary current at standstill. The curve O P F K is an arc of a 



















216 


ALTERNATING CURRENT MOTORS. 


circle having its center on the line O N prolonged. O L is the 
“ speed field ” current (assumed constant in the diagram, but 
properly accounted for in the computations). L M is the 
“ transformer field ” current, while O M is the total primary 
current at synchronism. 

The line F I drawn perpendicular to M T I represents the 
total loss of the machine at standstill—the proper scale being 
used. HI indicates the so-called “ constant ” losses, while 
F H shows the sum of the “ added ” primary and secondary 
copper losses. If the distance G H be laid off to represent 
accurately the easily determinable “ added ” primary copper 
loss, then F G shows the “ added ” secondary copper loss. 
Straight lines being drawn to join the point F and the point G 
with O of Fig. 56, if from any point P on the circular arc O P F K 



Fig. 56. —Current Locus of Single-phase Induction Motor. 


a perpendicular be dropped to the line M I the following values 
may be taken at once from the diagram: 

O P is the “ added ” component of the primary current, 
OP is also the “ added ” component of the secondary 

current,-- 

PL is the total secondary current, 

PM is the total primary current, 

Cos E M P is the power factor, 

P T -r- P M is the power factor, 

5 T is the “ constant ” losses of the machine, 

R S is the “ added ” primary copper loss, 

R T is the total primary losses (including “ speed ” field 
excitation current loss in secondary), 












TRANSFORMER FEATURES OF INDUCTION MOTOR. 117 


Q R is the “ added ” secondary copper loss, 

Q T is the total losses of the machine, 

P T is the input to the machine, 

Q P is the output, 

Q P -5- P T is the efficiency, 

PR is the total input to the secondary (excluding the 

speed field ” excitation current loss). 

(PQ + P R)% is the speed with synchronism as unity, 

(P QXP R)$ is the torque in synchronous watts. 

The representation of each of the quantities listed above, 
with the exception of the speed and the torque, will be appre¬ 
ciated at once from a comparison of Fig. 55b with Fig. 56, 
combined with a review of Chapter V. 

Speed and Torque of the Single-Phase Induction Motor. 

The speed and the torque can be ascertained in an extremely 
simple manner as follows: The “ speed ” field is under any 
chosen condition equal to the “ transformer ” field multiplied 
by the speed. Now the torque is proportional to the product 
of the “ speed ” field and that component of the secondary 
current which is in time phase with it and which crosses the 
core at the same mechanical position along the air gap as that 
occupied by the “ speed ” field. This component of the sec¬ 
ondary current is in time quadrature with the “ transformer ” 
field, and it has a value such that its product with the primary 
e.m.f. (for a unity ratio machine) represents the total power 
received by the secondary—exclusive of the loss due to the 
secondary excitation current. A little study will show, there¬ 
fore, that if the “ speed field ” were equal to the “ transformer 
field,” the torque in “ synchronous watts ” would be equal to 
the secondary input (excluding the excitation loss). Since the 
“ speed field ” varies directly with the speed, it is seen at once 
that the torque, D, is equal to the secondary input, W s , multi¬ 
plied by the speed, S. Thus, 

D = S W s . (1) 

The torque is also equal to the output W Q , divided by the 
speed, therefore 

D = W 0 + S 


hence 


5 = (IF 0 + W,)* 


( 2 ) 

(3) 


118 


ALTERNATING CURRENT MOTORS. 


That is to say, in a single-phase induction motor the speed is 
equal to the square root of the secondary efficiency. When the 
speed varies only a few per cent, from synchronism the slip is 
equal to one-half of the secondary loss expressed in per cent., 
as was pointed out by Mr. B. A. Behrend on page 884 of the 
issue of the Electrical World and Engineer for Dec. 8, 1900. 
Thus at a speed of .98 the secondary efficiency is .9604; the 
slip is .02; the loss is .0396. It is interesting to observe in this 
connection that in a polyphase induction motor the speed is 
equal directly to the secondary efficiency. 

Combining equations (1) and (3) above, it is found that the 
torque has the following value: 

D = (W 0 XW s )i (4) 

That is to say, the torque is equal to (P QXP R)* from 
Fig. 56, as used above. 

It is especially worthy of note that the speed and torque 
as here determined are not affected by the substitution of the 
modified circuits of Fig. 55b for the more nearly exact circuits 
of Fig. 55a. The method here outlined gives correct results 
both at synchronism and at standstill, and at other intermediate 
speeds the slight errors introduced are of both positive and 
negative values, and they tend to cancel in the final results. 

On account of the fact that at speeds below synchronism 
there is a slight decrease in the “ transformer-field ” current 
and a large decrease in the “ speed-field ” current (as it reacts 
upon the primary), while Fig. 56 assumes both of these currents 
to be constant, the operating power factor of a single-phase 
motor is somewhat greater than that shown in Fig. 57. The 
discrepancy is appreciable only in those cases where the “ syn¬ 
chronous no load ” current is large in comparison with the 
starting current. Thus Fig. 57 gives the power factor accu¬ 
rately for large motors, but small motors will show better power 
factors than there indicated. 

It is instructive to compare the performance of a certain 
polyphase motor, when operated normally, with that of the 
same machine when used as a 'single-phase motor. Using 
“ equivalent single-phase ” quantities throughout, the polyphase 
starting current of the motor, whose single-phase circle dia¬ 
gram is shown by the arc O P F K in Fig. 56, would be repre- 


TRANSFORMER FEATURES OF INDUCTION MOTOR. 119 

sented by the line M F F' (not completely drawn) having a 
length equal to twice that of the line M F (not drawn). The 
polyphase current locus is a circle passing through F' and 0, 
its center being on the line 0 N prolonged. If the machine is 
operated as a single-phase motor at a certain primary current, 
such as shown by M P in Fig. 56, the output is P Q, as noted 
above. If the same output is to be obtained when the machine 
is operated polyphase, then the equivalent single-phase value 
of the polyphase current will be M P' (not drawn), the line P P' 
being (practically) parallel with the line 0 F. Thus the volt- 
amperes input as a single-phase machine is greater than as a 
polyphase machine in the ratio of M P to M P' and the power 
factor is less in the ratio of Cos E M P to Cos EM P'\ the 
losses are also greater. 

Capacities of Single-Phase and Polyphase Motors. 

Although the circular diagram as developed above is ap¬ 
plicable to all types of single-phase induction motors, the com¬ 
parison just made refers exclusively to polyphase motors and 
the same motors used on single-phase circuits. The comparison 
between single-phase and polyphase machines is not quite so 
unfavorable to the former when each machine is designed pri¬ 
marily for its particular work. When a polyphase motor is 
operated as a single-phase machine, only a portion of the pri¬ 
mary copper is fully employed; evidently a greater output can 
be obtained by altering the inter-connections of the coils so 
as to use all of the copper. 

With an induction motor having uniformly distributed coils, 
when the iron is subjected to the same magnetic density and 
frequency, and the same current density is used in the copper, 
the output varies largely with the groupings of the coils. Thus 
it may be shown that with such a motor the volt-ampere input 
can be represented, relatively, by the periphery of a polygon 
having sides equal in number to the number of groups per .pair 
of poles, of which polygon the circumscribing circle represents 
the volt-ampere input for infinite groups, and double the diam¬ 
eter represents the input to the single-phase motor. Giving to 
the diameter of the circumscribing circle an arbitrary value of 
unity, the inputs to the machine for various groupings of coils 
are as follows: 


120 


ALTERNATING CURRENT MOTORS. 


Number of Groups. 
2 

3 

4 

6 


Type of Machine. 
Single-phase 
I'hree-phase 
Four-phase 
(Two-phase) 
Six-phase 
(Three-phase) 


Volt-Ampere Input 
2.000 
2.598 
2.828 

3.000 


Since the coils of commercial two-phase induction motors 
are grouped similarly to those of a four-phase machine and 
the coils of a three-phase motor are arranged similarly to those 
of a six-phase machine, a three-phase motor has a volt-ampere 
input 1.061 times that of a two-phase motor (on the basis of 
equality of losses), while the volt-ampere input of a single-phase 
motor is .707 times that of an equivalent two-phase machine. 
The facts upon which these statements are based are discussed 
more fully in the next chapter. 


CHAPTER IX. 


MAGNETIC FIELD IN INDUCTION MOTORS. 


Polyphase Motors. 

In construction an induction motor possesses as primary 
windings, coils placed mechanically around a core and sepa¬ 
rated as to polarization effects by the same number of angular 
degrees as the currents, which flow in the individual coils, 
differ in electrical time degrees, a two-pole model being assumed. 
Thus in a two-phase (or quarter-phase) machine the coils would 
be located 90 degrees one from the other, and in a three-phase 
motor the angular spacing would be 60 degrees (120 degrees). 

Consider a two-polar quarter-phase machine upon the sepa¬ 
rate windings of which there are impressed e.m.fs. in time 
quadrature. The e.m.f. at each coil demands that at each 
instant the resultant magnetism threading that coil have a 
certain definite value such that its rate of change generates in 
the coil the proper value of counter e.m.f., just as is true in any 
stationary transformer. No action which takes place within 
the machine can rob the primary coils of this transformer 
feature. Assume that the flux (and e.m.f.) follows a sine law 
of change of value with reference to time,and let $ be the max¬ 
imum value of flux demanded by the e.m.f. of each coil. Then 
at a given instant the e.m.f. in coils 1 and 2 expressed in c.g.s. 
units will be, for N effective turns. 


, T d d> . 

= N sin co t 
a t 


and 



d 4> 
d t 


sin 




hence 


e x = N co (j) cos co t 

e 2 = N co cf) sin cot or at 

any certain time t, the flux demanded by the e.m.fs. must have 

121 


\ 




122 


ALTERNATING CURRENT MOTORS. 


a value of <56 cos a> t threading coil 1 , and ^ sin cot threading 
coil 2 . 

In commercial induction motors the coils of one phase winding 
overlap those of the other, each winding being distributed over 
an area inversely proportional to the number of primary phases. 
The distributed character of the windings is such that the flux 
which threads one winding simultaneously threads the other, 
the distribution of the flux alone determining the actual effective 
value threading each coil at each instant. This feature of the 
winding combined with a slight value of modifying current in 
the closed conductors of the secondary winding is such that 
at any time t the fluxes in the two phase motor core have values 
< f i> cos c 0 t and <j> sin co t so distributed as to give a resultant of 

Vc f > 2 cos 2 co t+cf) 2 sin 2 co t = cp. 

That is to say, the resultant core flux is constant in value, 
but varies in position, producing the so-called revolving field. 
This field travels at a speed termed “ synchronous,” as found 
by the ratio of alternations per unit time to the number of 
poles of the machine. 

Within this revolving field is placed the secondary winding 
which in most cases is closed upon itself either directly or through 
certain variable resistance. When the rotor is traveling at syn¬ 
chronous speed, the secondary conductors are not subjected to 
change in flux except to the slight extent due to the irregulari¬ 
ties of the magnetic field, so that at this speed only the modify¬ 
ing current previously referred to flows in the secondary, and 
this current tends to cause the revolving field to have a truly 
constant value. 

In studying the internal actions of the induction motor, it 
is helpful to consider that there exists a revolving field of a 
certain value traveling at synchronous speed; that this field 
cuts the primary conductors at synchronous speed and at a rate 
to generate therein the proper counter e.m.f. and that the 
secondary conductors cut across this field at a rate depending 
upon the difference in the speeds of the rotor and of the revolv¬ 
ing field, that is upon the “ slip ” from synchronism. Ac¬ 
cording to this conception the effective value of the e.m.f. 
counter generated in each conductor on the face of the primary 
core will be the same irrespective of its mechanical position, 



MAGNETIC FIELD. 


123 


but the time-phase position of this e.m.f. will vary directly with 
the location on the core. When a number of conductors are 
connected in series to form a primary coil, the effective value 
of the resultant e.m.f. counter generated therein will be the 
vector sum of the e.m.f. of the individual conductors. From 
this fact may be determined the core flux necessary to give a 
certain counter e.m.f. with a certain distribution of the primary 
conductors. 

It is the purpose of the present chapter to discuss the facts 
upon which these statements are based, to show that the space 
distribution of the revolving magnetic flux follows a sine law, 
and to outline the method by which the implied considerations 
can be utilized in the treatment of induction motor phenomena. 

In explanation of several of the terms used below it should be 
stated that, in dealing with the magnetic field in induction 
motors of the multipolar type, it is necessary to distinguish be¬ 
tween “ electrical-time ” degrees, “ electrical-space ” degrees 
and “ mechanical-space ” degrees. Thus, in a four-pole machine, 
90 electrical-space degrees correspond to 45 mechanical-space 
degrees. Two fluxes may be said to be in electrical-time quad¬ 
rature when one reaches its maximum 90 time degree (J cycle) 
before or after the other. Independent entirely of their time- 
phase position, they would be in electrical space quadrature if 
their mechanical positions in the air-gap of a four-pole machine 
were displaced 45 mechanical-space degrees one from the other. 

Numerous writers when discussing the flux in the air-gap cf 
induction motors have stated in substance or have implied by 
their method of treatment that the space distribution of the 
magnetism depends largely upon the number of slots per pole 
per phase, and that the approximate sine wave found for the 
space-value of the flux is to be attributed to the subdivision of 
the coils of each pole winding. In order to lay emphasis on the 
fact that the space distribution of the magnetism is largely 
independent of the distribution of the'primary coils, and to 
lend simplicity to the explanations, the initial treatment below 
will be based on the assumption of coils of each phase being 
placed in a single slot per pole, giving the maximum of con¬ 
centration. The modifications which the distribution of the 
coils may introduce in the value of the flux, will later be treated 
in the simplest possible manner. It is well at this point to make 


124 


ALTERNATING CURRENT MOTORS. 


note of the fact that the effective value of the flux threading each 
primary coil is determined solely by the primary e.m.f., but that 
the space distribution of the flux depends upon the demands 
of the secondary circuits. 

Magnetic Distribution with Open Secondary. 

Figs. 57 to 77 indicate the behavior of the magnetism in the 
separate phase windings, and throughout the air-gap of a two- 
phase induction motor both when the secondary is on open cir¬ 
cuit and when it is on closed circuit with the rotor running at 
synchronous speed. It is well to investigate first the action 
which takes place in a single pole winding of one phase. The 
effective number of magnetic lines threading this winding is 
determined by the voltage impressed upon the coil and upon the 
number of alternations of the e.m.f. At each instant the re¬ 
sultant magnetism must have a value such that its rate of change 
generates in the coil the proper counter e.m.f., and for con¬ 
stant frequency this value is quite independent of any condition 
other than the pressure alone. A variation in the reluctance 
of the path of the lines alters only the magnetomotive force 
necessary to produce the lines. This magnetomotive force is 
supplied by current through the windings, having a value such 
that the ampere turns produce the required magnetomotive 
force. 

If the individual coils of the second phase of the two-phase 
primary winding, which are located on the core 90 electrical 
space degrees from those of the other phase, be subjected to the 
same conditions as assumed for the first phase, obviously the 
same results will be obtained. If the coils of both phases be 
subjected simultaneously to equal effective pressures at an elec¬ 
trical-time displacement of 90 degrees from each other, operat¬ 
ing conditions for a two-phase motor will be obtained. 

At each instant the rate of change of interlinking lines and 
turns per pole for the windings of each phase will be quite in¬ 
dependent of any condition other than the instantaneous value 
of the pressure of that phase, so that the presence of the wind¬ 
ings of the second phase can in no manner alter the effective 
value of interlinkages of the first phase winding, though a change 
in distribution of actual lines may occur. 

When the windings of one phase alone are energized, the core 


MAGNETIC FIELD . 


125 


magnetism evidently reaches its zero value once for each alter¬ 
nation of the pressure. With the pressure simultaneously active 
on the second phase winding, when the effective magnetism 




Figs. 57-63.—Flux De- Figs. 64-70.—Resultant Figs. 71-77.—Result- 
manded by each Phase; Core Flux; Secondary ant Core Flux; Second- 
Secondary open. Open. ary Closed. 


which threads the coils of the first phase is at its zero value, the 
magnetism threading the other phase is at a maximum. Now, 
since the coils of both phases occupy the same core and, in fact, 
















































































































































































































126 


ALTERNATING CURRENT MOTORS. 


the windings of each phase in effect surround the whole polar 
area of the core, the condition of zero effective magnetism for 
one phase winding can be accounted for only by the fact that 
each pole winding of that phase surrounds an equal number of 
north and of south lines. See Figs. 57 and 63. 

In intermediate conditions of the magnetism, the number of 
effective lines surrounded by the windings of each phase depends 
upon the slope of the e.m.f. curve of each phase considered 
separately. If the electromotive force of each phase follows a 
sine curve of time-value, the relative number of effective lines 
surrounded by one phase will vary as the product of the max¬ 
imum lines into the cosine of the angle of time, while those of 
the other phase at the same instant will vary as the sine of the 
same angle. 

The relative changes in the values of the magnetism threading 
each coil of a two-phase motor at 0, 15, 30, 45, 60, 75, and 90 
time degrees (during one-quarter cycle) are shown graphically 
in Figs. 57, 58, 59, 60, 61, 62, and 63, respectively. The values 
indicated are those which would be found for each primary phase 
winding operating singly, the secondary being on open circuit. 
Figs. 64, 65, 66, 67, 68, 69, and 70, respectively, show the re¬ 
sultant value and distribution of the core flux when the two- 
phase windings are subjected simultaneously to electromotive 
forces in time quadrature. By comparing, say Fig. 59 with 
Fig. 66, it will be seen that the effective value of the flux thread¬ 
ing each coil of one phase is in no way altered by the presence 
of the flux demanded by the e.m.f. impressed on the other phase 
winding. It is to be noted particularly that the flux produced 
by a current of any value whatsoever in one-phase winding has 
absolutely no effect upon the interlinkage of flux with the coils 
of the other phase. It is noteworthy also in this connection 
that the magnetomotive force necessary to produce a certain 
effective flux in one-phase winding t is neither increased nor de¬ 
creased by the presence of the flux due to the magnetomotive 
force of the other phase winding. That is to say, with the 
secondary on open circuit, each phase winding operates as 
though the other were not present, and as though it alone 
occupied the core. 

A comparison of Fig. 67 with Figs. 64 and 70 and the inter¬ 
mediate Figs. 65, 66, 68, and 69, will reveal the fact that. 


MAGNETIC FIELD. 


127 


when the secondary is on open circuit, the resultant core flux 
changes in mechanical position along the core, it varies in space 
distribution, and alters both in magnetic density and in total 
existing magnetic lines. Thus in Fig. 67, the flux is distributed 
over only one-half of the core area, where the magnetic density 
is %/2 times that indicated in Figs. 64 and 70, and the total 
number of magnetic lines is only v /,5 times that shown in either 
Fig. 64 or Fig. 70. It is seen, therefore, that under the condition 
here assumed, giving separately to the density and to the total 
flux in Fig. 67 or Fig. 70, the arbitrary value 100, the density 
of the core flux varies from 100 to 141.4 and the total number 
of lines existing on the core varies from 100 to 70.7 four times 
during each cycle. 

By noting the electrical space positions of the resultant core 
flux in Figs. 64, 67, and 70, it will be seen that the flux moves 
along the air-gap 45 electrical space degrees during 45 electrical 
time degrees, and that it advances a total of 90 electrical space 
degrees during 90 electrical time degrees. Thus, even when the 
secondary is on open circuit it travels around the air-gap at a 
certain definite speed termed “ synchronous,” although it 
varies in value and in space distribution. 

Magnetic Distribution with Closed Secondary. 

Assume, now, that the rotor is driven by some external means 
so that it travels at exactly synchronous speed. If the secondary 
conductors are on open circuit, electromotive forces will be gen¬ 
erated in them locally by the rate of change of the synchron¬ 
ously moving core flux. If now the secondary circuits be closed 
the electromotive forces generated by the changing core flux 
will produce currents in the secondary, which tend to prevent 
the variation in the magnetism which links with the secondary 
conductors. 

If the secondary circuits are thoroughly distributed over the 
secondary core, and are of perfect conductivity, then the most 
minute change in the value or space distribution of the synchron¬ 
ously moving magnetism will produce an enormous secondary 
current tending to maintain both the distribution and the value 
of the flux. As stated previously, the e.m.f. across each primary 
coil demands that a certain effective value of core flux at each 
instant threads through that coil, but the requirements of the 


128 


ALTERNATING CURRENT MOTORS. 


e.m.f. are met as fully with one space distribution of the flux 
as with another, so long as the effective value remains the same* 

The exacting requirements of the secondary conductors and of the 
electromotive forces impressed upon the phase windings of the pri¬ 
mary coils are completely fulfilled when the core flux assumes a sine 
curve of electrical space distribution , as shown in Figs. 71 to 77, 
inclusive. The proof of this statement is given below. 

A comparison of the sine curve of Fig. 71 with the rectangular 
curve of Fig. 64 will serve to determine the value of the max¬ 
imum ordinate of the sine curve. Since the area included be¬ 
tween each curve and its base line must be the same in the 
two cases, the maximum ordinate of the sine curve must be 

— times the maximum ordinate in the curve of Fig. 64, as will 

be verified incidentally below. 

A comparison of Fig. 6/ with Fig. 64 will show at a glance 
that the area enclosed by the former curve is much smaller than 
that enclosed by the latter, and the question naturally arises as 
to the possibility of any curve which surrounds a constant area 
serving to meet the demands for effective areas which differ so 
widely as do those indicated by the curves of Figs. 64 and 67. 

It will be observed that in connection with the sine curves of 
constant value but variable position shown in Figs. 71 to 77, 
inclusive, there are given also rectangular curves of constant 
position but variable value, which, as will be seen from Figs. 
57 to 63, inclusive, indicate the demand for effective area 
(magnetism) made by the e.m.f. of phase A. It remains now 
to be demonstrated that between the two points M and N 
which are constant in position, there is intercepted from the 
area represented by the sine curve as it moves along synchron¬ 
ously, an effective area equal in magnitude at all times to the 
area represented by the rectangular curve. 

Referring now to Fig. 71 let 

h = maximum ordinate of sine curve, 

then 

y = h sin x is the equation of the sine curve. 

The average ordinate of the whole curve from M to N (n ra¬ 
dians) is 

h C K • j h V 71 * 2h - /in 

— I sin x d x = — — cos x = — (1) 

ttJ 0 * lo * 


MAGNETIC FIELD. 


129 


Since the area of the sine curve is equal to that of the rectan¬ 
gular curve in Fig. 71, the maximum ordinate of the sine curve 


is equal to — times that of the rectangular curve, as mentioned 


previously. 

In Fig. 74, let the distance that the zero ordinates of the 
curve have traveled from M and from N be represented by a. 

Then the area below the base line will have the numerical value 
of 



h [ - cos a - ( - cos 0)] = h (1 - cos a) (2) 

This area will in effect neutralize an equal positive area, so that 
the remaining effective area will be 



sin x d x = 


2 h — 2 h (1 — cos a) = 2 h cos a 


0 ) 


This equation shows that the effective area bounded between 

« 

the ordinates at M and N and the sine curve is proportional 
directly to the cosine of the angle of displacement of the curve 
from the position giving the maximum area, as was assumed 
above. 

The interpretation of the above equation is that the effective 
area bounded by the sine curve and the ordinates M and N in 
Figs. 71 to 77 inclusive is in each case equal to the area repre¬ 
sented by the indicated rectangular curve. That is to say, 
magnetism of constant magnitude and distributed according to a 
sine curve of electrical space value will, when traveling syn¬ 
chronously around the air-gap, generate within each coil an 
electromotive force having a sine wave of electrical-time value. 
The relative electrical-time-phase position of the electromotive 
forces of coils distributed around the air-gap will depend solely 
upon the electrical-space positions of the coils. If independent 
sets of coils are located at intervals of 90 electrical-space degrees, 
the electromotive forces generated therein will vary from each 
other by \ period, and the coils may be interconnected to form 


130 ALTERNATING CURRENT MOTORS. 

a four-phase motor, or the opposite phase windings of this four- 
phase machine may be joined so as to form the familiar two- 
phase induction motor. Similarly, if independent sets of coils 
are located at intervals of 60 electrical-space degrees, the elec¬ 
tromotive forces generated therein will vary from each other 
by J period and the coils may be interconnected to form a six- 
phase motor, or the opposite phase windings of this six-phase 
machine may be joined so as to form the familiar three-phase 

induction motor. 

Determination of Core Flux. 

The determination of the value of the core flux can be based 
either upon the fundamental transformer equation or upon the 
equation used with alternating-current generators. Let 
n = number of turns in series per primary coil. 

E = effective value of primary e.m.f. per coil. 
f = frequency in cycles per second. 

From transformer relations 


E = 


V~2 

10 * 


f n.(f> 


m 


(4) 


where <fi m is the maximum value of the total flux threading the 
coil. 

Let A = the total air-gap area covered by the coil. 

When the secondary is on open circuit, and only one phase 

winding is active, 

o = = . _ 1Q8 —_ (5) 

m A \/2 ft f n A 

where B m is the maximum magnetic density at any point along 
the air-gap. (See Fig. 57.) 

When the secondary is on open circuit and both phase wind¬ 
ings are active. (See Fig. 67.) 

R _ \/2 <j) m _ 1Q8 (6) 

Bm ~ ~A~ ” ftfn A K ' 

When the secondary circuit is completely closed and the rotor 
is running at synchronous speed. (See Figs. 71 to 77.) 

ft 4> i 


B 


m 


10 s E 


2 A 2 \/ 2 j n A 


(7) 








MAGNETIC FIELD. 


131 


Treating the machine now as an alternator having n turns in 
series on the armature, with a flux of <j> m total lines per pole, 


E = 


V2- 

10 8 


f VI (£>m 


( 8 ) 


and the maximum magnetic density is, as found above, 



rr cf) m _ 10 s E 

2 A 2 y /2 f n A 


(9) 


The proof of the identity of the equations derived from trans¬ 
former and from alternator relations as given above, has been 
based upon the assumption of sine curves of electromotive 
forces. It is evident that, since the effective magnetism thread¬ 
ing each coil must vary at each instant according to the instan¬ 
taneous value of the e.m.f., when the e.m.f. wave is distorted the 
core flux must likewise vary from a sine curve of electrical space 
distribution. It is an interesting conclusion, which permits of 
easy verification that the mechanical distribution of the core flux 
follows a wave of electrical-space value similar in all respects to 
the electrical-time value of the primary electromotive force. This 
fact will be appreciated immediately if one considers that the 
instantaneous value of the e.m.f. generated in each armature 
conductor depends directly on the local magnetic density of the 
field through which it is moving at that instant. 


Effect on Core Flux of Using Distributed Winding. 

The problem of determining the effect of distributing the 
windings of each phase over a certain portion of the air-gap 
instead of concentrating them in one slot per pole, as assumed 
above, is rendered extremely simple by treating the machine as. 
an alternator, as was intimated in the opening paragraphs of 
this chapter. If the conductors which cross the face of the core 
and are joined in series to form a primary coil of one phase, are 
distributed over ft electrical-space degrees, then the resultant 
e.m.f. for a certain core magnetism is less than the arithmetical 
sum of the individual e.m.f. of the several conductors in the 
ratio of the chord of angle to the arc of the same angle. This 
result follows directly from the fact that the individual e.m.fs. 
are not in phase one with the other and it is necessary to take 
the vector sum of them. 




132 


ALTERNATING CURRENT MOTORS. 


In a two-phase motor (the equivalent of a four-phase ma¬ 
chine) the angle /? is 

angle /? = 90 degrees 

arc of/? = ^ 


chord of /? = \/2 


Therefore, in a two-phase motor, the maximum magnetic den¬ 
sity may be expressed as 


B 


m 


tc 10 8 E 

2 \/2 ' 2 \/ 2 " f nA 


7T 10 s E 
8 ' f n A 


( 10 ) 


In a three-phase motor (the equivalent of a six-phase machine) 
the angle /? is 

angle /? = 60 degrees 


arc of /? = 


7T 

3 


chord of p = 1 


Therefore, in a three-phase motor the maximum magnetic den¬ 
sity may be expressed as 


B 


m 



_10 8 E 

2\/2 / n A 


7T 10 8 E 

6 \/ 2 / n A 


( 11 ) 


Both equation (10) and equation (11) have been derived on 
the basis of the initial assumption that each turn of each coil 
spans an arc of 180 electrical space degrees. It is evident that 
if each turn covers an area less than that indicated by an arc 
of 180 degrees, the magnetic density must have a value greater 
than that given by these equations. In commercial induction 
motors one side of each coil is placed in the bottom of a certain 
slot and the return side of the same coil is placed in the top of 
another slot, with an arc of less than 180 electrical space degrees 
between the slots. 

Let the span of each coil be y electrical-space degrees, then 
the e.m.f. generated in the one side of each coil will be y elec¬ 
trical-time degrees out of phase with the e.m.f. in the other 
side of the same coil. If e is the e.m.f. in one side of a coil, 
the resultant e.m.f. of the coil will be 


E c — \/2 e \/1 + cos (180 — y) 


(12) 









MAGNETIC FIELD. 


133 


When y = 180 equation (12) reduces to 

E c = 2 e 

Hence, in general, for a two-phase motor 

D _ 7i 10 8 E Vl + cos (180 — y) 
m ~ 8 ' Jn~A ' vf 


and for a three-phase motor, 

— 1 n 8 F _ 

B m = lO f - A ■ Vl+COS (180 — r) 

12 / n A. 


(13) 


(14) 


(15) 


The last two equations refer to the magnetic density imme¬ 
diately at the bottom of the teeth of the primary core. The 
local magnetic density in the air-gap will depend upon the 
relative size of the slots and the teeth, and will be greater than 
that shown by these equations. 

Effect on Capacity of Varying the Grouping of Coils. 

An examination of the formation of equation (10), (11), (14), 
and (15) will reveal the fact that if in a certain induction motor 
the maximum value of the magnetic density is to remain con¬ 
stant while the coils are interconnected in different ways, the 
e.m.f. of each group of coils may be represented relatively as 
the cord of the arc in electrical space degrees which is covered 
by the coils in the group. It follows therefore that if one-half 
of the coils are connected continuously in series the total e.m.f. 
of the group of n coils in each of which there is an e.m.f. of e 

2 ti 

volts will be — e volts. If this value of volts be taken as 

7T 

unity, for the sake of comparison, then when one-third of the 
coils are joined in continuous series the total voltage of the group 
will be .866 volts. Likewise a group containing one-fourth of 
the coils would have a voltage of .707, and a group containing 
one-sixth of the coils would have a voltage of .500. 

If now it be assumed that each coil is to carry the same 
current as the other coils, then the volt-amperes per group 
will vary directly with the voltage. In consequence of this 
fact the total volt-amperes of an induction motor when oper¬ 
ated at constant maximum magnetic density in the core and 







134 


ALTERNATING CURRENT MOTORS. 


constant current density in the coils will be as follows for various 
groupings of the coils, assuming unit current: 

Number of 

Voltage 

Total 

Type of 

Groups. 

per Group. 

Volt-amperes. 

Machine. 

2 

1.000 

2.000 

Single-phase 

3 

.866 

2.598 

Three-phase 

4 

.707 

2.828 

Quarter-phase 

6 

.500 

3.000 

Six-phase 

(Three-phase) 


The relations shown in the above table were commented on 
in the preceding chapter. It will be noted that the volt-amperes 
rating of a single-phase motor are ./0/ times that of a Quarter- 
phase machine. A little study will show that a few of the pri¬ 
mary coils of each group may be removed without seriously 
decreasing the volt-amperes of the single-phase machine. Thus 
it is possible to materially decrease the primary copper without 


a proportionate 
following table: 

decrease in the 

volt-amperes, as 

i shown by the 

Percentage of 

Percentage of 

Saving in 

Decrease in 

Coils. 

Volt-amperes 

Copper 

Input 

100.00 

100.000 

.00 

.00 

88.89 

98.48 

11.11 

1.52 

77.78 

93.97 

22.22 

6.03 

66.67 

86.60 

33.33 

13.40 

55.56 

76.60 

44.44 

23.40 

50.00 

70.70 

50.00 

29.30 

It is seen from the above that a portion of the 

primary copper 


could be removed and yet the performance of the machine would 
be only slightly affected. It might seem that this fact would 
permit of a considerable saving in material, but a motor thus 
constructed would not in general be capable of being rendered 
self-starting from its primary circuits. It is the usual practice 
therefore to wind the primary completely and to use only a por¬ 
tion of the coils during normal operation, all of the coils being 
employed during the starting period. Commercial single-phase 
induction motors are frequently constructed as uniformly-wound 
three-phase machines, or as unsymmetrically wound two-phase 
machines. In the latter case the “ main ” winding contains 


MAGNETIC FIELD 


135 


about twice as much copper as the “ starting ” winding, and it 
occupies two-thirds of the core slots. 

Exciting Watts in Induction Motors. 

In the treatment above, the part played by the primary cur¬ 
rent in producing the revolving field has been practically neg¬ 
lected. It is well in this connection to show how the value of 
the exciting current may be determined directly from the 
volume of the air-gap and the volume of the core material, 
without reference to the required magnetomotive force, the 
number or the distribution of the primary coils. 

In Fig. 78a, let 

A = area of magnetic path, in sq. cm. 

I = length of path in iron, in cm. 



Fig. 78a. —Simple Magnetic Circuit. 


pL = permeability of iron. 
d = length of path in air. 
n = number of turns of coil. 

E = effective value of impressed e.m.f., in volts, 
c p = any chosen value of flux. 

i = any chosen value of exciting current, in amperes. 
I q = effective value of exciting current. 

<j) m = maximum value of flux. 

From fundamental magnetic relations 

m.m.f. 

UX reluctance 


4 > = 


4 tz n i 



10 


( 16 ) 

























136 


ALTERNATING CURRENT MOTORS 


As is well known, the reluctance of commercial magnetic 
material is not constant for all densities, and hence it is not 
proper to assume that the exciting current is sinusoidal when 
the flux is sinusoidal. When iron is included in the magnetic 
path, the exciting current wave will be peaked. When the 
major portion of the reluctance of the path is in air, the effect 
of the distortion produced by the presence of the variable re¬ 
luctance of the iron will not in general be very marked, and 
for all practical purposes it may well be neglected. 

Thus, if the maximum value of the exciting current is i m , 
the effective value will be slightly different from x/,5 brd 
since, in any event, the actual value of i m cannot be predeter¬ 
mined with a high degree of accuracy, due to the fact that the 
true value of /i is not known, it is safe to assume that for in¬ 
duction motors no measurable error is introduced by representing 
the effective value of the exciting current by the equation. 


But 


Then 


< fitn 

~A 



h1 = V.5 i 

m 




(17) 

$ m 

4 7T - T 

“ 10 

n 7 

(d 

A 

+ 

A 


(18) 



\ 


p.) 




10 (j) m | 

7 

|d + 

/> 

T > 

) 


(19) 


4 71 \/2 n A 





V 2 7 in} (j) 
* “ 10 8 

m 



(20) 

(quadrature) exciting 

watts may 

be expressed 

as 

W q = 

2 5 f 

| 

A 1 

^ 1 
+ 

) 

(21) 

= Al = 

volume of iron. 





= A d = 

volume of air. 

X 




fc+l) 

- (d A + 

/ A\ 


B 2 

LJ yyi 

(v a+ v A 

(22) 

\ / 

A 2 \ 

T ) 



\ fi / 

w q 

g 2 j 

2Q8 \ 

[v a + 

Vj 

) 


(23) 


and 








MAGNETIC FIELD. 


137 


where B m is the maximum magnetic density, in lines per square 
centimeter. 

The interpretation of equation (23) is, that in order to ascer¬ 
tain the (quadrature) exciting watts it is necessary to know 
only the maximum magnetic density, the volume and the per¬ 
meability of the iron, and the volume of the air-gap. That is 
to say, the very quantities which are necessary in order to deter¬ 
mine the core losses, will serve simultaneously jor the determination 
oj the ( quadrature ) exciting watts, when the permeability of the 
core is known. 

Although the erratic behavior of iron with reference to the 
change in its permeability cannot be reduced to a mathematical 
expression, it is found that for most practical purposes the per¬ 
meability of the iron used in transformers and induction motors 



Fig. 78b. —Composite Magnetic Circuit. 


may be expressed with a fair degree of accuracy, throughout the 
range of density from B m = 0 to B m = 15,000, by the equation 


H = 2,800- 3.2 


(7,500 - B m y 
10 5 


(24) 


Although the proof given above for equation (23) refers pri¬ 
marily to the magnetic circuits represented in Fig. 78a, it can be 
shown that the facts stated in connection with equation (23) 
apply equally as well to the circuits indicated in Fig. 78b and to 
the more complex circuits existing in both single-phase and 
polyphase motors. 

Thus, making use of proper subscripts, in Fig. 73b, 


















































138 


ALTERNATING CURRENT MOTORS. 


It may be shown theoretically and verified experimentally, 
that the (quadrature) exciting watts of a certain polyphase 
motor are the same in value when all phase windings are used or 
when only one winding is subjected to the primary pressure. 
Thus, when one phase winding of a two-phase motor is open- 
circuited, the other winding immediately takes double its former 
value of (quadrature) exciting current, the (quadrature) excit¬ 
ing watts remaining the same. These facts are discussed more 
fully below. 

It seems, therefore, that the most logical way to determine 
the exciting current of an induction motor is to ascertain the 
density of the magnetism in the core and in the air-gap, and then 
calculate the quadrature watts, just as one ordinarily calculates 
the core loss watts. 

Magnetic Field in the Single-Phase Induction Motor. 

As was mentioned in a previous chapter, when the circuits 
of a polyphase induction motor operating near synchronism 
are so arranged as to convert the machine into a single-phase 
motor, the revolving field, which was previously due to the 
combined actions of certain components of the displaced poly¬ 
phase currents, continues to exist, and the action of the machine 
in developing mechanical power is subjected to almost no change. 
It is equally well known that the quadrature “ speed ” com¬ 
ponent of the magnetic field is produced by current in the sec¬ 
ondary, and that the magnetomotive force represented by this 
secondary current must be supplied by a component of pri- 
marv current. On account of the fact that the secondary 
current which produces the quadrature “ speed field” occupies 
a position in space such that it cannot possibly react directly 
on the field produced by the primary current, it is not immedi¬ 
ately apparent in what manner the “ quadrature watts ” for 
the “ speed field ” are supplied by the primary exciting mag¬ 
netomotive force. 

A popular method of treating the internal behavior of a 
single-phase induction motor is the one due to Ferraris, who 
showed that the simple alternating field can, in all of its effects, 
be replaced by two revolving fields moving in opposite directions, 
the maximum value of each being equal to one-half of the maxi¬ 
mum value of the alternating field. By means of this method 

( 

\ i 


MAGNETIC FIELD . 


139 


it is possible to ascertain the distribution of current in the 
secondary, and the reaction of certain components of the sec¬ 
ondary current upon the primary, but the present writer be¬ 
lieves that the actual significance of the results obtained, as 
viewed by the average reader, are greatly obscured by the 
difficulty in distinguishing the imaginary from the real when 
the two are so closely interwoven. 

A prominent writer of the present day states that “ the 
cause of the cross magnetization in the single-phase induction 
motor near synchronism, is that the induced armature currents 
lag 90° behind the inducing magnetism and are carried by the 
synchronous rotation 90° in space before reaching their max¬ 
imum and that “below synchronism the induced armature 
currents are carried less than 90°, and thus the cross magneti¬ 
zation due to them is correspondingly reduced and becomes 
zero at standstill.” It is greatly to be doubted if these state¬ 
ments convey any physical idea whatever to a mind not already 
thoroughly familiar with the facts. 

Although the method outlined below will not serve to present 
any facts which cannot be ascertained by other methods which 
have frequently been employed, yet it is believed that 
much good can be accomplished by drawing attention to the 
fact that all of the phenomena connected with the production 
of the magnetic field in the single-phase induction motor can 
be investigated with the utmost simplicity by dealing directly 
with well-known electro-magnetic relation without resorting 
to imaginary physical or mathematical representations. This 
method has been touched upon in a preceding chapter. It is 
believed that a more extended discussion thereof is desirable 
at this point. 

Production of Speed-Field Current. 

Fig. 79 shows a two-pole model of a single-phase induction 
motor which is represented as possessing four mechanical poles, 
two of which (1 and 3) are excited by single-phase alternating 
current. Merely for sake of simplicity in explanation, mechanical 
poles 2 and 4 are indicated as subjected exclusively to the flux 
of the “ speed field.” Under any condition of operation the 
flux in poles 1 and 3 is determined directly by the primary 
e-m.f., modified by the volts consumed in the local impedance 


140 


ALTERNATING CURRENT MOTORS. 


of the primary coil by the current which flows therethrough. 
That is to say, the flux in these poles follows the laws which 
relate to stationary transformers; no action which takes place 
in the secondary can rob the primary of this transformer fea¬ 
ture. In dealing with the secondary, however, it is necessary 
to recognize the fact that each rotor conductor is subjected 
to four distinct electromotive forces—the e.m.f. ! s produced by 
the rate of change of the transformer and of the speed fields, 
and the e.m.f’s generated by the motion of the rotor through 
the transformer and speed fields. Each of these e.m.fs will 



be treated separately and the combined effects will then be 
investigated. 

When the rotor is moving across the transformer field in 
the direction indicated, there will be generated in each of the 
conductors under the poles an e.m.f. proportional to the pro¬ 
duct of the field magnetism and the speed of the rotor. Evi¬ 
dently if the speed be constant, of whatsoever value, this e.m.f. 
will vary directly with the strength of magnetism; that is, 
will be maximum when the magnetism is maximum, and 
zero at zero magnetism. Other conditions remaining the same 
the maximum value of this secondary e.m.f. will vary directly 
with the speed of the rotor. 























MAGNETIC FIELD . 


141 


If the circuits of the rotor conductors be closed, there will 
tend to flow therein currents of strengths depending directly 
upon the e.m.f.’s generated in the conductors at that instant 
and inversely upon the impedance of the rotor conductors. 
The current which flows through the rotor circuits at once 
produces a magnetic flux which by its rate of change in value 
generates in the rotor conductors a counter e.m.f. opposing the 
e.m.f. that causes the current to flow, and of such a value 
that the difference between it and this e.m.f. is just sufficient 
to cause to flow through the local impedance of the conductors 
a current whose magnetomotive force equals that necessary 
to drive the required lines of magnetism through the reluctance 
of their paths. Since this latter magnetism must have a rate of 
change equal (approximately) to the e.m.f..generated in the rotor 
conductors by their motion across the primary field, and since 
this e.m.f. is in time phase with the primary field, it follows 
that this magnetism must have a value proportional to the rate 
of change of the primary magnetism; that is, it is (approxi¬ 
mately) in time quadrature to the primary magnetism. 

A study of the direction of the currents in the rotor under the 
conditions assumed will show that when a north pole at 1 (in 
Fig. 79) has reached its maximum value and is decreasing to¬ 
wards zero, the speed field is building up with a north pole at 
2, and that this pole continues to increase in strength until the 
magnetism at 1 reverses its direction. Thus, it may be stated 
that the north pole of the resultant magnetism travels in the 
direction of motion of the rotor. Since the rapidity of reversal 
in sign of the “ transformer field ” poles and of the “ speed field ” 
poles depends solely upon the frequency, it may be stated that 
the resultant field revolves at synchronous speed. The “ speed 
field ” is equal (approximately) to the product of the “ trans¬ 
former field ” and the speed, with synchronism as unity. Thus 
the resultant field is at any speed elliptical as to electrical space 
representation; one axis of the ellipse is determined by the 
“ transformer field,” while the other depends upon the speed. 
At synchronism the ellipse becomes a circle; above synchronism 
the ellipse has its major axis along the “ speed field ”; at zero 
speed the ellipse is a straight line, which means that at standstill 
there is no “ space ” quadrature flux and hence no revolving 
field. 


142 


ALTERNATING CURRENT MOTORS. 


Reviewing the electromagnetic processes just discussed, it 
will be noted that the e.m.f. which produces the “ speed field ” 
current is caused by the motion of the rotor through the “ trans¬ 
former field ” and is opposed by the rate of change of the “ speed 
field ” through the rotor circuits. The mechanical position of 
the “ speed field ” current with reference to the primary coil 
prevents it from reacting directly on the “ transformer field.” 
It remains to investigate the effect of the e.m.f’s generated in 
the secondary by the rate of change of the “ transformer field 
through the rotor conductors and by the motion of the rotor 
conductors through the “ speed field.” It will be noted at once 




Figs. 80a and 80b. —Production of Transformer Secondary 
Currents and Electromotive Forces. 


that a current due to either of these e.m.f’s would be in position 
to tend to affect the “ transformer field.” 

Transformer Features of the Single-Phase Induction 

Motor. 

Referring now to Fig. SOa assume initially for sake of simplicity 
that the rotor revolves at absolutely synchronous speed (being 
driven by some external means). As noted above, the trans¬ 
former e.m.f. of the “ speed field ” in the rotor is slightly less 
than the speed e.m.f. of the “ transformer field ,” and is out of 
time phase therewith, by an amount equal to the e.m.f. necessary 

























MAGNETIC FIELD. 


143 


to cause the “ speed field ” current to flow through the “ local 
impedance of the rotor conductors. It is well to establish 
the equality between the actual “ speed field ” current and 
the component of secondary current which reacts upon the 
transformer field. 

When the speed of the rotor is exactly synchronous, the trans¬ 
former e.m.f. of the “ speed field ” in the rotor (see Fig. 80a) is 
exactly equal in effective value to the speed e.m.f. of the speed 
field, and is in exact time quadrature therewith, assuming 
sinusoidal time values for the flux. Likewise the transformer 
e.m.f. of the “ transformer field ” in the rotor is exactly equal 
in effective value to the speed e.m.f. of the transformer field, 
and is in exact time quadrature therewith. It is evident, 
therefore, that if the vector difference between the speed e.m.f. 
of the transformer field and the transformer e.m.f. of the speed 
field causes a certain value of current to flow through the local 
impedance (including resistance and local leakage reactance 
but not the counter e.m.f. effect of the speed field) of the rotor 
in a mechanical position to produce the magnetomotive force 
necessary to cause the magnetism of the speed field to flow 
through the reluctance of its path, an exactly equal current 
will flow through an exactly equal local impedance of the 
rotor in mechanical position to affect the transformer field— 
due to the vector difference between the transformer e.m.f. 
of the transformer field and the speed e.m.f. of the speed field. 
A change in the value of the local impedance of the rotor, by 
some alteration in its mechanical construction, may vary 
slightly the value and time-phase position of the speed field, 
but the equality in effective values of the two currents in elec¬ 
trical space quadrature and in electrical time quadrature in 
the rotor is not affected thereby. 

Analysis of the Rotor Electromotive Forces. 

On account of the confusion which is liable to be caused by 
any misconception of the various individual transformer and 
motor features of the single-phase induction motor, especially 
with reference to the source of the exciting magnetomotive force, 
it is desirable to point out definitely the several transformer 
and motor actions and to discuss their inter-relations. A pro¬ 
lific source of confusion exists in connection with the frequent 


143a 


ALTERNATING CURRENT MOTORS. 


though erroneous assumption that the e.m.f. required for forcing 
a certain value of current through the local rotor impedance 
in the “ speed field ” axis is different from that necessary to 
force an equal value of current through the local rotor impedance 
in the “ transformer field ” axis. This assumption is based on 
a lac 1, of distinction between the local rotor impedance and the 
self-t+ A”dive impedance of the rotor circuits. The latter quan¬ 
tity n c-udes the transformer effect of the “ speed field ” flux, 
while the former includes merely the effect of the local leakage 
flux and the resistance of the rotor circuits. Thus, the actual 
e.m.f. necessary to produce a certain value of current in the 
rotor in a direction to produce the “ speed field ” (that is, the 
“ speed e.m.f. of the transformer field ”) must overcome the 
self-inductive reactance due to the speed field flux in addition 
to the local rotor impedance. It is most convenient to consider 
the actual e.m.f. to consist of two components, one to over¬ 
come the self-inductive reactance and the other to overcome 
the local rotor impedance. The former, is the quantity desig¬ 
nated as the “ transformer e.m.f. of the speed field ” which (at 
synchronous speed) is exactly equal to the “ speed e.m.f. of the 
* speed field.” The vector difference between the actual “ speed 
e.m.f. of the transformer field ” and the “ transformer e.m.f. 
of the speed field ” is the e.m.f. which causes the actual speed 
field current to flow through the local rotor impedance ; an ex¬ 
actly equal e.m.f. (the vector difference between the “ trans¬ 
former e.m.f. of the transformer field ” and the “ speed e.m.f. 
of tne speed field ” at synchronous speed) causes an exactly 
equal current to flow through the exactly equal local rotor im¬ 
pedance in a mechanical position to tend to affect the trans¬ 
former field. 

Even if one assumes the local rotor impedance to be in¬ 
definitely decreased and the “ speed ” field more and more to 
approach in value the “ transformer ” field (at synchronous 
speed) the exact value of the actual “ speed field ” exciting 
current will be only inappreciably increased, while the equality 
between this current and the current which directly opposes 
the “ transformer ” field will not be altered in the least. The 
limit would be reached when both the local rotor impedance 
and the e.m.f. reduce to zero. In this case zero e.m.f. divided 
by zero impedance gives a value exactly equal to the “ speed 


MAGNETIC FIELD. 


143b 


field exciting current. These statements can readily be 
verified by a little study of Fig. 80a. 

From the facts discussed above it is evident that a current 
exactly equal to the 14 speed field ” current is produced in the 
rotor in an electrical space position such that its magnetomotive 
force tends directly to affect the “ transformer field.” Since 
the “ transformer field ” must have the value demanded by the 
primary e.m.f., a current equal in magnetomotive force and 
opposite in direction to this component of the secondary current 
must flow in the primary coil. As indicated in Fig. 80a, and 
as may be verified by a study of the fluxes and currents, this 
component of the secondary current has a time phase position 
to tend to decrease the “ transformer field,” so that the op¬ 
posing current in the primary appears as an added component of 
the primary exciting current. Thus the “ speed field ” current 
is accurately represented in the exciting magnetomotive force 
supplied by the primary current. 

It is interesting to note that the “ added ” component of the 
primary exciting current depends upon the reluctance of the 
path taken by the flux of the “speed field”; when the air 
gap traversed by the “ speed field ” is much greater than that 
through which the “ transformer field ” passes (as shown in 
Figs. 79 and 80a), the “ added ” component is likewise much 
greater than the true primary “ transformer ” exciting current. 
Thus the total quadrature exciting watts are equal to the sum 
of the watts which would be required for producing the same 
magnetic field by means of two-phase currents in coils wound 
symmetrically on poles 1, 2, 3 and 4, and not necessarily equal 
to twice the value taken by the windings on poles 1 and 3, with 
the secondary circuit open. 


Secondary Currents in the Single-Phase Motor. 

It is instructive to investigate the conditions which would 
exist if the two components of secondary current at synchronous 
speed could be caused to continue to flow unaltered with the 
primary on open circuit. As noted above, the “ speed field ” 
current and that component of the secondary current which 
tends to oppose the transformer field flow in “ electrical time 
quadrature ” and occupy positions in “ electrical space quad- 


144 


ALTERNATING CURRENT MOTORS. 


rature thus, if acting without opposition, they would produce 
a rotating magnetic field. It is a curious fact, easily appreciated 
from a study of Figs. 79 and 80a, that this magnetic field would 
travel around the air-gap in a direction opposite to the motion 
of the rotor. Since the two exciting components of the sec¬ 
ondary current in reality combine in the rotor structure to 
produce a resultant single current distributed throughout the 
several conductors, it may be stated that a band of secondary 
exciting current revolves synchronously in a negative direction. 
If one considers the time value of the current in a single rotor 
conductor, he will discover that at synchronous speed this cur¬ 
rent is of double frequency. As will be shown below, at 
other speeds the “ secondary exciting current ” has a value 
proportional (approximately) to the speed, and it continues to 
revolve synchronously in a negative direction; thus the fre¬ 
quency of this current in an individual rotor conductor is equal 
to the primary frequency, f P , multiplied by one plus the speed, 
S, with synchronism as unity. That is, f s = } p (1 + 5). 

Consider now the effect of operating the rotor at a speed 
somewhat below synchronism. Since there is no opposing 
magnetomotive force in line with the “ speed field ” the “ speed 
field ” component of the rotor current acts as though it alone occu¬ 
pied the secondary conductors, and its value is in no way affected 
by the presence of any other component of secondary current. 
Thus, the e.m.f. necessary to force the “ speed field ” current 
through the secondary conductors depends solely on the value of 
the “ speed field ” component of the rotor current. Since the 
e.m.f. generated in the secondary by the motion of the conductors 
through the “ transformer field ” depends directly upon the 
product of this field and the speed, it follows that a definite 
percentage of this speed-generated e.m.f. is consumed in the 
“ local ” secondary impedance at all speeds, and that the time 
phase displacement between the speed-generated e.m.f. and the 
transformer e.m.f. of the “ speed field ” is constant at all times. 
Thus, the “ speed field ” at speed, 5, bears to the “ transformer 
field ” a ratio equal to the product of 5 and a certain constant 
which denotes the difference in value and phase position of the 
“ speed field ” and the “ transformer field ” at exact synchronism. 
The significance of this statement is that the “ speed field ” 
component of the secondary current has a value proportional 


MAGNETIC FIELD. 


145 


accurately to the product of the speed, S, the “ speed field ” 
current at synchronism and the ratio of the ‘ transformer field '* 
at speed, 5, to that at synchronous speed. 

Since the transformer e.m.f. of the “ transformer ” field in 
the rotor depends upon the strength of this field, but is independ¬ 
ent of the rotor speed, while the opposing speed e.m.f. of the 
“ speed field ” varies with the product of the speed and the 
“ speed field ” it follows that the resultant e.m.f. which tends 
to produce power ’ current in the secondary at speed, 5, is 
equal (approximately) to the product of the quantity (1—5 2 ) 
and the transformer e.m.f. (See Fig. 80b.) This component of 
secondary cuirent occupies at all times a space position mag¬ 
netically in line with the “ transformer field,” and it reacts 
upon the primary just as though it flowed through the secondary 
of a stationary transformer into a non-inductive (fictitious) 
load resistance; it is superposed in space, but not in time, upon, 
that component of the “ revolving secondary exciting current ” 
which directly opposes the “ transformer field.” 

In Fig. 80b let the line 0 A represent the value of the trans¬ 
former e.m.f., E t , of the “transformer field ” in the rotor; E t 
varies directly with the “ transformer field,” and hence decreases 
as the primary current increases. Let the angle AO B represent 
the time phase difference between E t and E s , the speed e.m.f. 
of the “ speed field ” in the rotor;, the angle A O B is constant 
at all speeds. At synchronous speed, E s has a value O C such 
that the resultant of E t and E s gives the electromotive force, 
E r , which produces that component of rotor current which re¬ 
acts upon the primary. At some lower speed, 5, E s has a value 
O C v such that O C x = S 2 (0 C), neglecting the relative decrease 
in the value of E t , and the resultant electromotive force which 
produces current to react upon the primary is shown by A C v 
Of this latter e.m.f. the component, C t D v in time quadrature 
with the transformer e.m.f., varies with S 2 , the square of the 
speed; (that is, it decreases when S decreases), while the com¬ 
ponent, A D v in time phase with the transformer e.m.f., varies 
with (1—S 2 ); that is, it increases with decrease of speed. To 
the latter of these components may be attributed the secondary 
“ load ” current, while to the former may be attributed that 
component of the “ negatively revolving exciting current ” 
which directly opposes the “ transformer field.” 


146 


ALTERNATING CURRENT MOTORS. 


When the rotor is stationary the “ load ” component of the 
secondary current in the individual conductors is of the pri¬ 
mary frequency, at nearly synchronous speed it pulsates in 
value in each separate rotor conductor, being unidirectional in 
certain conductors and alternating at double frequency in cer¬ 
tain other conductors situated 90 electrical space degrees from 
the former. 

It is seen, therefore, that there exist in the rotor three com¬ 
ponents of secondary current, each of the primary frequency 
with reference to space representation: the “ speed field ” cur¬ 
rent, the current having a value closely equal to the product 
of the “ speed field ” current, and the speed, but displaced 
therefrom both in space and in time by 90 electrical degrees, 
and the load current. Each of these varies in value with the 
“ transformer field.” The first varies directly with the speed 
5. The electromotive force which produces the second varies 
with S 2 , while the electromotive force which produces the third 
varies with (1—5 2 ). At synchronous speed the first two com¬ 
ponents are equal in value, while the third is practically zero. 
At zero speed the first two components are zero and only the 
third flows in the rotor. Under all conditions the second and 
third components combine to form the secondary current of 
the machine considered as a transformer, the second compo¬ 
nent acts as a continually decreasing (with decrease of speed) 
wattless current, while the third acts in all respects as though 
it flowed through the secondary into a non-inductive load re¬ 
sistance. 

Graphical Representation of Secondary Quantities. 

The relations which exist between the several components 
of the fluxes, the currents and the electromotive forces in the 
rotor at various speeds are shown graphically in Figs. 81a and 81b. 
In Fig. 81a, let the distance, A D, be given an arbitrary value 
of unity, and let the curve, A E F D, be a semi-circle. Then 
if D E is made equal to the speed, 5, B D is equal to S 2 . Con¬ 
sider the condition when the speed, S, has the value represented 
by F D\ the ratio of the “ speed field ” to the “ transformer 
field ” is shown directly by the line, F D; this line also shows 
the ratio of the true “ speed field ” current to the true 
“ transformer field ” current, and likewise the ratio of the 


MAGNETIC FIELD. 


147 


component of the secondary current which directly opposes 
the “ transformer field ” to the true “ speed field ” current. 
Thus, if at the speed shown by F D, A D is assumed equal 
to the “ transformer field ” current, D F is equal to the true 
“ speed field ” current and CD is equal to the “ opposing ” 
component of the secondary current. Furthermore if at 
the speed, F D,A D be made equal to the e.m.f. which 
would be produced in the secondary by the “ transformer 
field ” with the rotor stationary, C D is the actual speed e.m.f. 
due to the motion through the “ speed field,” and A C is the 
e.m.f. which causes “ load ” current to flow through the sec¬ 
ondary impedance. 


G 




F IG . 81 a. —Numerical value of Fig. 813 . —Time and Space Values 
Currents and Electromotive Forces. of Fluxes and Currents. 

It is to be noted especially that the diagram of Fig. 81a gives 
only the relative numerical values of the various components 
and does not indicate their time-phase or electrical space po¬ 
sitions. The electrical space and time values of the electro¬ 
motive forces are shown in Fig. 80b, while the equivalent values 
for the fluxes and currents are represented in Fig. 81b. In this 
diagram G H is equal to A D of Fig. 81a, while the curve, GLHI y 
is a circle; J K is made equal to F D and the curve, G K H J y 
is an ellipse; M N is equal to CD and curve, M K N J, is an 
ellipse. The electrical space value of the flux at synchronous 
speed is shown by circle, GLFL1 while at speed, D F, it has 
the value indicated by ellipse, GKHJ. If the line, G H, 









148 


ALTERNATING CURRENT MOTORS 


shows the value and phase position of the true transformer 
field ” current, the line, J K , simultaneously shows the value 
and phase position of the true “ speed-field ” current; these cur¬ 
rents are in separate electrical structures, and they do not com¬ 
bine directly, but their magnetomotive forces combine to pro¬ 
duce the elliptical revolving magnetic field. The value and 
phase position of the “ opposing ” component of the secondary 
current is shown by the line, N M ; this current is in the same 
electrical structure with the current, J K, and the two combine 
to produce the “ negatively revolving secondary exciting cur¬ 
rent,” shown by curve, K M J N, which is elliptical as to space 
representation. 

It is interesting to observe, that the actual “ speed field ” cur¬ 
rent in the secondary varies directly with the speed, but that 
the component of the secondary current which reacts directly 
upon the transformer field varies with the square of the speed, 
or, more correctly, with the square of the transformer field. 
It will be noted that on account of this fact the total “ quad¬ 
rature exciting watts ” of the single-phase induction motor 
vary directly with the square of the “ transformer field ” plus 
the square of the “ speed field.” Thus the true exciting watts 
of the machine at any speed are directly proportional to the 
sum of the squares of the densities of the fluxes traversing the 
several magnetic paths, as was mentioned previously. 


CHAPTER X. 


SYNCHRONOUS MOTORS AND CONVERTERS. 

Synchronous Commutating Machines. 

The term “ synchronous commutating machines ” refers to 
all motors or generators which receive or deliver both alter¬ 
nating and direct current. The machines discussed below are 
rotary converters and double-current generators, and compari¬ 
sons are made with the capacities of alternating-current gen¬ 
erators or motors of different number of phases. 

For simplicity in treatment, the rotary converters are as¬ 
sumed to deliver at the direct-current commutator all of the 
power received at the alternating end; that is, the output is 
assumed equal to the input in determining the relative currents 
on each side, though, as will be seen later, the armature copper 
loss is properly accounted for. The double-current generators 
are assumed to deliver equal amounts of power at the commu¬ 
tator and at the collector rings. The assumption is further 
made that the alternating-current wave in each case follows a 
true sine curve of time-value. 

In a rotary converter the mean flow of alternating current is 
in a direction opposed to the flow of the direct current, but the 
absolute value of the alternating current varies from time to 
time and the direct current reverses direction of flow through 
the individual coils as each passes under one of the brushes, 
so that the resultant current in the coils varies both in value 
and direction of flow from instant to instant and, in general, 
it has not the same heating effect in different armature coils. 

When the alternating current has unity power factor, the 
maximum value of the current, evidently, occurs when the 
group of coils of the phase under consideration are developing 
their maximum -e.m.f. The mechanical position of the coils at 
this instant of maximum e.m.f. is that in which the center of 
the group of coils is passing at right angles to the lines of force 
from a field pole—with a non-distorted field this position 

149 


150 


ALTERNATING CURRENT MOTORS. 


would be opposite the center of the field pole. W hen the coils 
are passing parallel to the lines of force the e.m.f. is of course 
zero. At intermediate positions, the value of the e.m.f. may 
be represented by E m cos d, where E m represents the maximum 
e.m.f. and 0 the angle between the instantaneous position of 
the coils and a line from the pole center to the center of the 
armature shaft. 

The absolute value of the maximum e.m.f. depends upon the 
number of coils in the group considered. While the effective 
value of the e.m.f. developed in each coil is the same as that 
in the others, and adjacent coils are connected in series, the 
effective value of the e.m.f. of a group of coils is not proportional 




Figs. 82a and 82b. —Phase Relations of Voltages. 

directly to the number of coils composing a group, since the 
e.m.f. of one coil is not directly in phase with that of the adja¬ 
cent coils; that is, the e.m.f. of each coil reaches its maximum 
value at a different instant from that corresponding to the 
maximum e.m.f. of each of the other coils. 

If time be represented as angular degrees passed over by the 
armature of a bipolar machine, and the value of the e.m.f. of 
each individual coil be denoted by a line of any chosen length, 
and the line for each coil be placed in the angular-time position 
which the armature would occupy when that coil has its max¬ 
imum e.m.f., a diagram similar to that represented by Fig. 82a 
will be produced. Here 01 represents the effective value and 
time position of the e.m.f. in coil No. 1, and 02 represents 










MOTORS AND CONVERTERS. 


151 


corresponding quantities for coil No. 2, etc. As stated above, 
these coils are connected in series, so that the actual effective 
value of the e.m.f. of the coils as interconnected may be repre¬ 
sented as in Fig. 82b.- It will be observed that as the number 
of coils is increased the figure approaches a circle and that in 
any case the extremities of the sides lie on a circle. 

By the use of Fig. 82b or the equivalent circle it is a 
simple matter to determine the effective value of the e.m.f. of 
a group of coils on an armature. This e.m.f. is seen to be 
represented in value by the chord of the arc subtended by the 
group of coils. If the total number of coils on the armature 
be divided into P equal parts, then the angle covered by each 
part is 360° s-P\ and, since the chord is equal to twice the sine 
of half the angle, the e.m.f. of each group is 

„ n E . 180° . 180° 

E P = 2 — sin — = E sin —p~ 


where E is the value of the e.m.f. measured across a diameter. 
Now, E is the effective value of the e.m.f. at the diameter, 
while for a rotary converter the direct-current commutated e.m.f. 
is equal to the maximum value of this e.m.f., or is v^2 E = E m . 
Therefore, the effective maximum value of the e.m.f. of 


a group of 
to 


coils which cover -p- part of the armature is equal 


E m . 180° 

— sin —= E p . 


v 7 2 


P 


When the alternating current is in phase with the e.m.f., 
the product of the current flowing in the coils selected, by the 
e.m.f. across the group gives the power in watts in that section 
of the armature and when the armature is symmetrically loaded, 
the total power is 

E m . 180° 

W — P I p —7= sm 

V2 P 


Synchronous Motors and Generators. 

For the purpose of subsequently comparing the capacities of 
alternating-current machines of various types and phases, it is 
convenient at this point to ascertain, by means of the above 
formula, the relative capacities of a closed-coil armature used 




152 


ALTERNATING CURRENT MOTORS. 


in a direct-current generator and the same armature used in 
alternating-current generators of different number of phases. 
Consider the armature to revolve in a field of constant intensity 
at a constant speed. There will be generated the same e.m.f. 
per conductor irrespective of the connections of the external 
circuits. Assume that the capacity is in each case wholly de¬ 
termined by the heating of the armature conductors and, as a 
method of direct comparison, assume that the external load is 
so adjusted in each case that there flows the same current through 
each conductor on the armature whether used in a direct or an 
alternating-current generator and independent of the number 
of phases. Obviously, the loss from heating of the armature 
conductors will always remain the same, while the capacity 
will vary as the external load. 

TABLE i. 


Capacities of Alternator Compared to Direct-Current Generator as 100. 


Number 

of 

Phases. 

(Rings.) 

P 

Volts 

Between 

Leads 

100 . 180 

V2 Sm P 

Ep 

Amperes 

Per 

Phase. 

Ip 

Total 

Output. 

P Ep Ip 

Wt 

Amperes 

Per 

Lead. 

2 Wt 

70.71 P 

II 

2 

70.71 

5 

707.1 

10.00 

3 

61.24 

5 

918.6 

8.66 

4 

50.00 

5 

1000.0 

7.07 

6 

35.35 

5 

1060.6 

5.00 

Infinite 

0.+ 

5 

1110.7 

0.+ 


For simplicity in comparison assume that there flows always 
5 amperes in each armature conductor, and also that the e.m.f. 
measured between the direct-current brushes is 100. The 
capacity as a direct-current generator is evidently 1000 watts, 
while the outputs as alternating-current generators of various 
numbers of phases will be as in Table I. (See also Fig. 83a.) 

When P = infinity, E P = 0, but P E P = 100 X\/A Xtt, as 

* “ 

will be shown later. 

It is interesting to note that the capacity of an alternating- 
current generator can be represented as the perimeter of a polygon 
having sides equal in number to the number of phases, of which 
polygon the circumscribing circle represents the capacity for 
infinite phases, double the diameter of this circle representing 








MOTORS AND CONVERTERS. 


153 


the capacity of the so-called single-phase generator, while the 
capacity of the machine as a direct-current generator is repre¬ 
sented in value by the perimeter of a square inscribed within 
the circle. These facts will be brought out by an inspection 
of Fig. 83a. 

It is to be observed that the output given above is the volt- 
ampere capacity of each machine. With an alternating-current 
generator, the power delivered will, of course, vary with the 
power factor. At any power factor less than unity the ratio of 
the alternating to the direct-current capacities would vary 



Closed-coil Generator Armature. 

directly therewith, but the ratio of the alternating-current 
capacities for different numbers of phases would remain the 
same independent of the power factor. 

In the equations given, P corresponds to the number of col¬ 
lector rings. Thus, a closed-coil single-phase generator, so- 
called, is considered a two-phase generator, the phases being 
180° apart. A so-called two-phase generator having a closed- 
coil armature with four collector rings is, in fact, a four-phase 
generator with phases 90° apart, though its capacity is neither 
increased nor decreased by loading as two separate two-phase 
(so-called single-phase) generators. 












154 


ALTERNATING CURRENT MOTORS. 


Synchronous Converters, Unity Power-Factor. 

The problem of determining the relative capacities of rotary 
converters and other synchronous commutating machines can 
be attacked by use of the same fundamental equation developed 
above as applied to the alternating-current generators, though 
the method of application must be slightly modified to suit 
the various types of machines. Perhaps the simplest method 
of ascertaining the effect of the presence of both the direct and 
the alternating current upon the relative copper loss of the 
armature is to compare the losses of different machines for 


TABLE II. 

Volts and Amperes for Same Power with Different Numbers of Phases. 


Number 

of 

Phases. 

P 

Volts 

Between 

Leads. 

100 . 180 

V2 Sm P 

= Ep 

Amp. per 
Phase; Ef¬ 
fective. 

W 

P. Ep 
= Ip 

Amp. per 
Phase; 

Max. 

\/2 Ip 

= Im 

„ Amp. per 
Lead; Ef¬ 
fective. 

v 7 2W 

P. E 

= lL 

2 

70.71 

7.071 

10.000 

14.142 

(single-phase) 

3 

61.24 

5.443 

7.698 

9.428 

4 

50.00 

5.000 

7.071 

7.071 

(two-phase) 

6 

35.35 

4.714 

6.665 

4.714 

Infinite 

0.+ 

4.501 

6.365 

0.+ 

D. C. 

App. 2 

E= 100 

1 = 5.0 

T 5 - 0 

1 = 10 

assumed 

equal outputs, 

and then 

to determine 

the relative 


outputs for the same loss. 

The formula referred to above enables one to determine at 
once the effective value of current which is necessary to give 
a certain amount of power when P, the number of phase, and 
Em, the direct e.m.f. are known. Table II gives the value of 
current for various number of phases for an assumed power of 
1000 watts and direct e.m.f. of 100 volts. 

It is to be. noted that as the number of phases increases the 
current per group of coils decreases, but that, even with an 
infinite number of phases, the current has yet a finite value. 







MOTORS AND CONVERTERS. 


155 


An inspection of Fig. 83a will show that the total e.m.f. of the 
infinity groups of infinity phases is represented by the circum¬ 
ference of the circumscribing circle. The value of current to 
produce the assumed power is found by dividing the 1000 
watts by this total e.m.f. 

The maximum value of current for sine waves is VT times 
the effective value and, when the power factor is unity, this 
maximum current flows when the coils are developing their 
maximum e.m.f. With an armature in a bipolar field, as shown 
in Fig. 83b, the maximum value of e.m.f. in a group of coils occurs 



Fig. 83b. —Current in Armature Conductors of Four-phase 
Synchronous Converter. 


at that position of the revolution of the armature where the 
line joining the extremities of the group is in a vertical plane, 
and the e.m.f. in other positions varies as the cosine of the angle 
of deviation from the vertical position. 

Having determined the value of the maximum alternating 
current and the position of the group of coils when this max¬ 
imum flows it now remains to investigate the effect of the 
presence of the direct current in the armature coils. 

For purpose of combined generality of treatment and sim¬ 
plicity of discussion, the so-called single-phase rotary will be 
omitted for the present and there will be discussed first the so- 





































156 


ALTERNATING CURRENT MOTORS. 


called two-phase machine which is in reality a four-phase 
rotary converter. Since there are four phases, the group of 
coils for each phase covers 90 degrees. Assume that there are 
72 coils on the armature. There will then be IS coils per phase, 
and each coil covers 5°. (The treatment here given is general 
and results will be in no way affected if the 5° contain any 
number of coils, or in fact, less than one coil.) 

Consider the instant when the group of coils is in the position 
at which the maximum e.m.f. is generated, as indicated in Fig. 
83b. The alternating current is equal to \/2 / = 7.071, while 
the direct current is 500^100 = 5, so that the actual current 
flowing through the coils is 7.071 — 5, causing a relative loss of 
(2.07) 2 Xl8 = 77, where the resistance of each coil is taken as 
unity. 

As the armature moves forward 5° the alternating current drops 
to 7.071 cos 5° = 7.05, while the direct current remains at 5, 
causing a relative loss of (2.05) 2 X18 = 76. In this manner 
the relative loss for each position of the armature may be de¬ 
termined up to that number of degrees rotation which brings 
the beginning of the group of coils under the + brush. When 
the armature has rotated 50° one coil of the group considered 
will be on the right of the brush, and, though this coil has the 
same value of alternating current in it as has each of the others 
of the group, the direct current through it is reversed and the 
resultant current is, therefore, greater than in the other coils 
or is equal to 4.55 + 5 = 9.55, causing a relative loss of (9.55) 2 
Xl = 91.2. The other coils have at this instant a resultant 
current of 4.55 — 5 and a relative loss of (—.45) 2 X17 = 3.4, 
hence the total relative loss for the group is 91.2 + 3.4 = 94.6. 

As the armature continues to rotate, more coils pass into the 
rieht-hand section and less remain in the left-hand section, till, 

o 

when the armature has rotated 90° from its initial position, the 
coils are equally divided between the two sections—one-half 
on each side of the brush. At this instant the alternating cur¬ 
rent will have decreased to zero and the total relative loss will 
be 450, which is the loss due to the direct current alone. 

Continuing this investigation till the armature has rotated 
180°, it will be plain that the conditions obtained at the begin¬ 
ning are being repeated, so that a mean of the total relative 
losses throughout the 180° is the same as occurs continuously 


MOTORS AND CONVERTERS. 


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158 


ALTERNATING CURRENT MOTORS. 


throughout the operation of the rotary. As shown by Table III, 
this mean relative loss is 170, for the conditions herein assumed. 
The relative loss for the group of coils if the machine were 
operating as a direct-current generator would be 450, as found 
above. The relation between these, 1 to 2.647, indicates the 
relative loss of the four-phase rotary converter compared with 
the corresponding direct-current generator, at the same output, 
as unity. Since the loss in any circuit varies as the square of 
the current, the relative currents to give the same loss should 
vary as the square root of the above ratio, or the relative ca- 

Time- Degrees from Point of Maximum E.M.F. and of Maximum Current 


10 20 30 10 50 60 70 80 00 100 110 120 130 140 150 160 170 180 190 200 210 220 230 



Fig. 84.— Distribution of Loss in Armature of Four-phase 
Synchronous Converter, the Angle of Lag Being Zero. 


pacities of the machine as a rotary and as a direct-current 

generator will be \/2.647 = 1-627. 

An inspection of columns 4 and 5 of Table III or of equivalent 
curves of Fig. 84 reveals the manner in which the instantaneous 
value of current in the coils varies. It will be seen that, though 
the mean effective value of current for the group of coils is less 
as a rotary than as a direct-current generator, there are certain 
coils which at certain times carry more current than others 
and that one coil will carry a maximum of twice the current 
when operating as a four-phase rotary as a direct-current 
generator. 




















































































































































































MOTORS AND CONVERTERS. 


159 


Distribution of Heat Loss in Armature Coils. 

As a result of the variation in strength of the alternating 
current at the instant when each separate armature coil of a 
rotary converter or a double-current generator passes under a 
commutating brush, at which time the direct current within 
the coil is reversed, the maximum value of current to which a 
coil is subjected varies with the individual coils according to 
the location of each within the group constituting the windings 
of one phase—the windings between two adjacent collector-ring 
taps in a bipolar armature. The two end-coils, that is, the coils 
which connect the alternating-current leads to the adjacent 
groups on either side, carry greater values of current than the 
contiguous coils within the group, but these two coils do not have 


TABLE IV. 


Type of 


Rotary 

Double-Current 

Machine. 


Converter. 


Generator. 

Number 

100% 

90.63% 

100% 

90.63% 

of 

Power 

Power 

Power 

Power 

Phases. 

Factor. 

Factor. 

Factor. 

Factor. 

2 

3.00 

3.21 

1 .50 

1 .60 

3 

2.33 

2.70 

1 .27 

1.35 

4 

2.00 

2.47 

1 .21 

1 .28 

6 

1 .67 

2.21 

1.17 

1 .24 

Infinite 

1 .00 

1 .60 

1.14 

1 .20 


equal current values when the power factor of the alternating 
current is less than unity. 

A knowledge of the relative increase in instantaneous value 
of maximum current is important and a study of its effect and 
location as to coils is instructive as indicating the existence of 
local heating within the armature windings. The value of this 
maximum current which flows within a single coil depends upon 
the number of phases and the power factor of the current. 
Table IV gives the relative values of this maximum current for 
different machines considering as unity the current in a direct- 
current generator at the same load. 

Although the alternating current follows a sine wave, the 
current in individual coils does not follow a sine curve of time- 
value, and compared to its effective heating value, the max¬ 
imum value is much greater than that obtained with a true • 





160 


ALTERNATING CURRENT MOTORS 


sine curve. The maximum current flows for only a small frac¬ 
tion of the total time and is confined to a relatively small por¬ 
tion of the armature, so that the excess heating effect cannot at 
once be judged from Table IV, but must be determined by 
calculation similar to those recorded in columns 11 and 12 of 
Table III. 

Table V indicates the relative values of the maximum and 
the minimum losses in individual coils on the armature of syn¬ 
chronous commutating machines of various phases at power 
factors of 100 per cent, and of 90.63 per cent., compared to the 
mean armature loss per coil under the same condition of service. 
The results here recorded, therefore, indicate the relative lack 
of uniformity of distribution of heat loss in the armature wind¬ 
ings. 

TABLE V. 


Maximum and Minimum Losses in Individual Coils. 


Type of 
Machine. 


Rotary Converter. 


Double-Current 

Generator. 

Number 

of 

Phases. 

100% 

Power Factor. 
Max. Min. 

90.63% 

Power Factor. 
Max. Min. 

100% 

Power Factor. 

Max. Min. 

90.63% 

Power Factor. 
Max. Min. 

2 

2.270 

.331 

2.462 

.334 

1.201 

.650 

1.462 .443 

3 

2.161 

.405 

2.748 

.343 

1 .084 

.828 

1.132 .647 

4 

1.926 

.531 

2.600 

.391 

1 .048 

.903 

1 .094 .753 

6 

1.590 

.725 

2.217 

.461 

1 .018 

.955 

1.065 .852 

Infinite 

1.000 

1 .000 

1 .000 

1.000 

1.000 

1 .000 

1 .000 1 .000 


Synchronous Converters, Fractional Power Factor. 

When a rotary converter is operated at a power factor less 
than unity two effects are observed: There is required a propor¬ 
tionately larger current to produce a given power, and the 
maximum current does not flow in a given group of coils when 
the coils are generating their maximum e.m.f. Though the 
relative loss for a given machine does not vary regularly with 
decrease in power factor, it is sufficient for present purposes to 
determine the effect of operating the machines at a single fairly 
low value of power factor. 

Assume that the current lags 25° behind the rotary e.m.f. 
The power factor is, therefore, cos 25° = .9063 and the max¬ 
imum value of current instead of being 7.071, as before for the 









MOTORS AND CONVERTERS. 


161 


four-phase rotary converter is now 7.071 -h. 9063 = 7.81 am¬ 
peres, and this maximum occurs when the armature has moved 
forward 25° from the position giving maximum e.m.f., or, what 
is the same thing, the armature must now rotate forward only 
20° before the group of coils begins to pass under the brush 
instead of 45°, as when the current and e.m.f. are in phase. 

Bearing these facts in mind and making proper substitution 
in Table III, there is obtained by a method similar to the one 
used previously a mean relative copper heat loss in the group 
of coils of 266 (Table VI) which, compared to the loss of 450 


Time- Degrees from Point Maximum E.M.F. 

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 

Degrees from Point of Maximum Current 

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 



20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

0 


600 

500 

400 

300 

200 

1<X> 

0 


Fig. 85. —Distribution of Loss in Armature of Four-phase 
Synchronous Converter at 25° Lag. 


for the direct-current generator, indicates a relative output of 

J 1^2 = 1 300 as compared with that of the direct-current gen- 
\ 266 

erator. 

Double Current Machines. 

The method of determining the output from the double-cur¬ 
rent generator is quite similar to that used above. In this 
case, however, the direction of flow of the alternating current 
at the time of maximum value is the same as that of the direct 





























































































162 


ALTERNATING CURRENT MOTORS 


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XX 
































MOTORS AND CONVERTERS. 


163 


current, so that the sum, and not the difference, must be taken 
in determining the resultant current. Columns 13 and 14 of 
Table III will show the manner in which the instantaneous value 
of resultant current varies. The relative loss of 1631.6 is for 
equal direct and alternating-current outputs so that the rela¬ 
tive total output for the same loss as in a direct-current gen¬ 
erator will be 



_450 
1631.6 


7TT— = 1.050. 


Table VI records the calculation for determining the effect 
of a lag of 25° In the alternating current for the double-current 
generator, and, as the method is quite the same as used for 
the rotary converter, it need not be further discussed. 


TABLE VII. 

Relative Capacities of Alternating-Current Machines; Direct-Current Capacity = 1. 


Type of 
Machine. 

Rotary Converter. 

Double-Current Gen. 

Alt.-Cur. 

Generator. 

Number 

of 

Phases. 

100% 

Power 

Factor. 

90.63% 

Power 

Factor. 

100% 

Power 

Factor. 

90.63% 

Power 

Factor. 

100% 

Power 

Factor. 

90.63% 

Power 

Factor. 

2 

.848 

.731 

.951 

.890 

.7071 

.6418 

3 

1 .338 

1.103 

1 .023 

.992 

.9186 

.8325 

4 

1.627 

1 .300 

1 .050 

1.020 

1.0000 

.9063 

6 

1.937 

1 .482 

1 .066 

1.038 

1.0610 

.9612 

Infinite 

2 291 

1 648 

1 .082 

1 .052 

1.1105 

1 .0066 


Relative Capacities of Synchronous Machines of Various 

Phases. 

The relative capacities of alternating-current machines com¬ 
pared with a direct-current generator as unity are given in Table 
VII. The results recorded in Table VII are plotted in the form 
of curves in Fig. 86, so as to show to the eye the effect of varying 
the number of phases of a given machine. It will be seen at a 
glance that increasing the number of phases in each case in¬ 
creases the capacity of the machine, but that the relative in¬ 
crease for alternating-current and double-current generators is 
small compared to the increase for rotary converters. A 
change from three to six phases with an alternating-current 
generator results in an increased capacity of 15.5 per cent., 






164 ' ALTERNATING CURRENT MOTORS. 

while an equivalent change with a rotary converter produces 
from 35 per cent, to 45 per cent, greater output, depending 
upon the power factor of operation. These latter figures may 
be increased or decreased if the current wave departs materially 
from the assumed sine curve of time-value, though the figures 
as given represent results obtained in practice. 



Fig. 86. —Capacities of Alternating Current Machines Com¬ 
pared to Direct-current Generator. 

The increase in capacity of a rotary converter resulting from 
a change from the so-called single-phase to three-phase is 
from 51 per cent., to 57 per cent. Therefore, changing a given 
rotary from three-phase to six results in from 70 per cent, to 
80 per cent, as great increase in capacity as changing from 
single to three-phase. Reference to Table V will show that 





















































































































































































MOTORS AND CONVERTERS. 


185 


with a three-phase rotary converter one section of the armature 
windings is subjected to from 5.3 to 8 times as great current 
heating effect as certain others, while, with a six-phase machine, 
the corresponding results are 2.2 and 4.8. This means that 
the heat loss is much better distributed in a six-phase con¬ 
verter armature than in a three-phase one. 

Since the transformers necessary to convert the three-phase 
current from the high potential transmission circuits to six-phase 
for rotaries are in no way more expensive or complicated than 
those for three-phase rotaries, economy in cost of equipment 
and efficiency of operation dictates the use of six-phase ma¬ 
chines, and where the size of the rotaries operated justified the 
additional connecting circuits between the transformers and the 
machines, six-phase rotary converters should be used. 

Characteristic Performance of Synchronous Converters. 

In external appearance a polyphase rotary converter resem¬ 
bles a direct-current generator with a conspicuously large com¬ 
mutator and an auxiliary set of collector rings. Its design in 
certain respects is a compromise between alternating-current 
and direct-current practice. This is most noticeable with 
reference to the speed and number of poles; that is, the fre¬ 
quency. A careful review of constructive data for modern 
direct-current railway generators reveals the fact that the 
frequency of such machines is between 8 and 12 cycles while 
the frequency of the older belted type was about 20 cycles. 
Alternating-current generators, on the contrary, when not lim¬ 
ited in frequency, are seldom built for less than 60 cycles. 

Since the rotary converter is in fact a synchronous motor, it 
must run at a speed determined by the alternations of the sup¬ 
ply and the number of its poles. A limit to the possible in¬ 
crease in speed of the converter is set by the peripheral speed 
of the commutator. Experience has demonstrated that 3000 
ft. per minute is as high as the commutator should run to give 
reliable service. The peripheral speed of a rotary converter is 
equal to the product of the number of alternations by the dis¬ 
tance between two adjacent neutral points. For a given direct- 
current e.m.f., a limiting potential difference of from 8 to 10 
volts between segments, and the minimum size of bars, allow¬ 
ing for insulation, it is evident that, with a limiting peripheral 


166 


ALTERNATING CURRENT MOTORS. 


speed, there is soon reached a limit to the number of alterna¬ 
tions. While, under the limits noted, it is possible to construct 
500-volt rotary converters for 60 cycles, good design has deter¬ 
mined that 25 cycles is the proper frequency for converter 
work at such pressure. 

Since the direct and alternating currents flow in the same 
windings, revolving in the same field, it will be appreciated that 
the direct-current voltage bears a constant ratio to the alter¬ 
nating voltage; the maximum value of the internal alternating 
e.m.f. being equal to the direct e.m.f. The effective value 
of the alternating e.m.f. observed will depend upon the 
form, of the e.m.f. wave and upon the points on the windings 
between which the voltage is taken. Assuming a sine wave 
and denoting the direct e.m.f. by 1, the effective alter¬ 
nating e.m.f. is about 0.71 for two-phase machines and star- 
connected six phasers, and about 0.61 for three-phase and delta- 
connected six-phase machines. The wave form, and hence this 
ratio, can be materially changed by altering the slope of the 
pole-faces. 

Excitation of Synchronous Machines. 

The counter e.m.f. of the converter, both as to wave form and 
magnitude, must be equal to that of the supply system. If the 
converter does not tend of itself to produce a wave similar 
and equal to that of the system, corrective currents will flow 
in the armature windings, which currents so react upon the 
field that the generated e.m.f. will have a wave form exactly 
the same as that of the supply. As far as converter output is 
concerned, these corrective currents are wattless. They, how¬ 
ever, affect the regulation of the system and waste energy in 
the resistance of the connecting circuits and should therefore 
be eliminated when possible. 

If the converter is excited to give an e.m.f. less than that of 
the system when it is running at the speed at which the alter¬ 
nations of the supply designate that it must run, a lagging cur¬ 
rent will be drawn from the system, which current tends to 
strengthen the motor field so that the generated e.m.f. is made 
equal to that of the system. Similarly, if the converter field 
is overexcited, the current drawn from the supply mains will 
be leading and will thus demagnetize the field sufficiently to 


MOTORS AND CONVERTERS. 


167 


make the e.m.f. equal to that of the system. It is thus plain 
that the current demanded depends upon the field excitation 
and will be least for that excitation which would cause the 
machine to generate an e.m.f., when running at normal speed, 
equal to that' of the system. 

Hunting of Synchronous Machines. 

The mean speed of the converter must equal the mean speed 
of the generator, but the instantaneous speeds of the two may 
be quite different, as will be seen later. The synchronizing 
current, which holds the converter in step with the system, 
tends to cause the converter to follow any irregularity in the 
frequency of the supply-current. 

The tendency to irregular angular velocity in each revolution 
is inherent in the construction of reciprocating engines and is 
augmented by the periodic hunting of the governors of engines 
operating alternators in parallel. The inertia of the converter 
armature causes it to tend to run at a constant speed, and if 
the alternations of the supply are irregular, during a portion 
of the time the converter will be ahead of the system, and at 
other times it will be lagging behind. During the time of rela¬ 
tive phase displacement between the converter armature and 
the system, the synchronizing current acts to draw the arma¬ 
ture into perfect step. If additional forces are brought to 
bear upon the converter during the period of phase shifting, 
the relative oscillations may be either increased or decreased 
according to the time-direction of such forces. 

This action is very similar to the swinging of a pendulum. 
If, when the swing is in one direction there is given it an im¬ 
pulse in the same direction, the amplitude of the swing is in¬ 
creased, and if the impulse is given in the opposite direction, 
the amplitude is diminished. The periodic hunting of the 
engine governor, the steam admissions, the momentum of the 
reciprocating parts, the inertia of the generator armature and 
that of the converter armature, are elements which tend to 
increase or diminish this oscillation. 

Much theoretical and experimental work was undergone be¬ 
fore a complete cure for the tendency to this periodic phase 
shifting was found. Among the methods at present used may 
be mentioned the heavy flywheel effect for the converter, and 


168 


ALTERNATING CURRENT MOTORS. 


a magnetically weak armature compared with the field. In 
either of these cases, the converter armature tends to revolve 
at a mean speed independent of the relative irregularity of 
the frequency of the supply. 

The method which has been found most satisfactory for the 
prevention of hunting of rotary converters is the use of damping 
devices. These are usually of the form of copper shields be¬ 
tween or surrounding the poles, often covering a portion of the 
pole-tip or even imbedded in the pole proper. As the armature 
oscillates back and forth across its normal position, the shifting 
armature magnetism, produced by the unconverted portion of 
the motor current, induces current in the low-resistance copper 
shields, which current always opposes the shifting magnetism 
producing it. The damping action thus brought into play 
when the field is suddenly distorted has the effect of suppressing 
the oscillations. When the alternations of the supply are ir¬ 
regular, the damping devices act to cause the converter to 
tend to follow the irregularities, but prevent an exaggeration 
of the momentary phase displacement of the armature and thus 
have a steadying effect upon the whole system. 

Starting of Synchronous Converters. 

Before a rotary can be placed into active service, it must be 
brought up to synchronous speed and into step with the supply 
system. Methods in use for accomplishing this result are as 
follows: 

(1) Since polyphase currents are universally used as supply, 
the application of the alternating currents directly to the sta¬ 
tionary armature without field excitation will result in a rotating 
magnetic field about the armature core. The eddy currents 
thereby induced in the pole-faces will exert a torque on the 
armature and cause it to tend to speed up to synchronism. 
Under the condition of starting, the step-up transformer rela¬ 
tion between the field and armature windings causes a rela¬ 
tively large e.m.f. to be generated in each field coil. To lessen 
danger from this source, the windings on the separate poles 
may be isolated from each other so that the e.m.f. generated 
in the coils will not be in the normal series relations, and thus 
the total e.m.f. across any two points may be limited to that 
generated in one pole winding alone. When a shunt to the 


MOTORS AND CONVERTERS. 


169 


series coils is used, it must be opened at starting, otherwise the 
heavy alternating current sent through it and the series coils 
may cause excessive heating. 

(2) Where the- station equipment will permit, the converter 
may be started up as a direct-current motor. The direct cur¬ 
rent may be obtained from a storage battery, from another 
converter or a motor-generator set may be installed for this 
purpose. A device for automatically tripping the direct-current 
circuit-breaker upon closing the alternating-current switch has 
proved a valuable addition to the equipment for converters 
started by this method 

(3) A method extensively employed by one of the leading 
manufacturing companies is the use of separate motors for starting 
one or more of the converters of the sub-station equipment. 
The motors may conveniently be of the induction type and 
therefore started by standard methods for this purpose. A 
common location for the induction motor secondary is upon 
an extension of the converter shaft. Since the induction motor 
must experience a slip of some value, it is necessary, in order 
to bring the converter to full synchronism, for the motor to 
have a less number of magnet poles than the converter. 

Compounding of Synchronous Converters. 

In street railway and similar work it is always desirable to 
increase the station pressure as the load comes on, in order 
that the line voltage shall remain more nearly constant. The 
dependence of the direct-current voltage of the converter upon 
that of the alternating supply has been commented upon. In 
order, therefore, to increase the pressure of the output it is 
necessary to increase the pressure of the supply also. This 
increase may be obtained by the use of variable-ratio step- 
down transformers, or by the insertion of reactance in the 
supply circuit and running the load current through a few 
turns around the field poles. 

We have found previously that if the converter is over¬ 
excited, leading currents will be drawn from the supply, while 
if the excitation is below normal, lagging currents will be drawn. 
If a lagging current be drawn through a reactance, the collector 
ring voltage will be lowered. If, however, leading current be 
drawn through the reactance the voltage will be raised. The 


170 


ALTERNATING CURRENT MOTORS. 


change in the phase of the current to the converter is governed, 
by the excitation, which is in turn regulated by the load current, 
so that, with series reactance, the effect of the series coils on 
the field of the converter is quite similar to that of the com¬ 
pounding on the ordinary direct-current railway generator. In 
operation it is sometimes found that the transmission line and 
converter circuits possess sufficient self induction so that addi¬ 
tional reactance is unnecessary. 

Inverted Converters. 

The rotary converter is an entirely reversible piece of appar¬ 
atus. If fed with alternating current of a certain voltage, it 
will supply direct current of a corresponding (not equal) volt¬ 
age, and similarly, if fed with direct current it will deliver 
alternating current of corresponding voltage. When operated 
to convert from direct to alternating current, the rotary is 
called by the somewhat ill-chosen term “ inverted converter.” 

When driven by alternating currents its speed is governed by 
the alternations of the supply quite independent of all other 
conditions. When run from the direct-current side, however, 
its speed is determined by the relation of its field strength and 
impressed e.m.f. at the brushes. It operates in this respect 
exactly like a direct-current motor. If the field from any 
cause becomes weakened, the converter will speed up until its 
armature conductors cut the field magnetism at a rate to gen¬ 
erate an e.m.f. equal to the internal impressed e.m.f. If the 
field be strengthened the speed will be correspondingly decreased. 
Obviously, therefore, the frequency of the alternating-current 
output may be quite irregular, though the e.m.f. be constant, 
if the field fluctuates in strength. 

As far as the alternating-current output is concerned, the in¬ 
verted converter operates as a generator. In an alternating- 
current generator, lagging currents weaken the field, while 
leading currents have the opposite effect. With constant ex¬ 
ternal field excitation, the running strength of the field will, 
therefore, depend upon the character of the alternating-current 
load. When used to supply power for induction motors and 
similar apparatus, the current drawn will have a lagging com¬ 
ponent which weakens the field and tends to increased speed. 
Safety to the converter and motors necessitates that the increase 


MOTORS AND CONVERTERS. 


171 


in speed be limited, while satisfactory service requires that this 
tendency be counter-balanced. 

The following methods are in use for overcoming the tendency 
to irregular speed: 

(1) Magnetically weak armature compared with the field. It 
is possible by this method to operate a converter on full zero 
power factor current without very materially weakening the field. 

(2) Separate field excitation supplied from a direct-current 
generator driven synchronously with the converter. Any in¬ 
crease in converter speed causes the exciter generator to supply 
more field current and thus counteracts the influence of the 
lagging armature current. This latter arrangement can be 
made to regulate for very small variations in speed by operating 
the exciter field below saturation and the converter field at a 
high magnetic density and having the converter armature 
relatively magnetically weak. A very slight increase in speed 
causes a large increase in field current, while at the same time the 
armature current has a small demagnetizing effect upon the field. 

The operation of a rotary converter is in general much more 
satisfactory than that of a corresponding direct-current gen¬ 
erator. This is due to several causes: (1) Absence of field dis¬ 
tortion; the rotary is both a generator and a motor. As a 
generator the armature current tends to distort the field in a 
direction opposite to the distortion as a motor. The effects of 
the armature reactions, therefore, neutralize each other, and since 
there is no shifting of the field, the point of commutation does 
not vary with the load, and sparkless commutation results. 
(2) Lessened friction loss. (3) Greater output from same arma¬ 
ture. Since the load current at portions of each revolution 
feeds directly from the alternating-current side without travers¬ 
ing the whole winding, as must be the case with a generator, the 
effective armature resistance is less than it is for the same 
armature used in a generator. 

Predetermination of Performance of Synchronous 

Converters. 

Due to the simultaneous operation of a rotary converter, 
both as a motor and as a generator, the field distortion from the 
motor action is to some extent counteracted by that from the 
generator action, as has just been stated, so that under proper 


172 


ALTERNATING CURRENT MOTORS 


field excitation, the field strength remains quite approximately 
constant throughout a great range of load. Hence, the armature 
iron loss varies but slightly with the load and, with a degree 
of accuracy fairly equivalent to that obtaining with constant 
potential-transformers, the iron loss may be considered to be 
independent of the load current. The variable loss is due almost 
exclusively to the copper loss in the armature winding. 

The rotary converter with constant impressed alternating 
e.m.f. considered as a direct-current generator, tends always to 
produce the same direct external e.m.f. The apparent measur¬ 
able pressure, however, drops off as the load is applied, due to 



the copper loss of the armature, and such drop is a direct measure 
of the loss within the armature. 

At any chosen value of load current the sum of this loss in 
watts added to the output watts of the converter gives a value 
which would be directly determined by the product of the direct 
e.m.f. at its no-load value and the load current at its chosen 
value. It thus appears that with load amperes plotted as ab¬ 
scissas and watts as ordinates the curve of armature output 
plus copper loss due to load current is a right line and may be 
drawn at once for any value of output current (Fig. 87). The 
ratio of the watts loss in the armature copper, due to any load 

































MOTORS AND CONVERTERS. 


173 


current, to the value of the load current gives the effective 
value of the armature resistance. 

Knowing the no-load losses of the converter and the effective 
armature resistance the complete performance may be calcu¬ 
lated as follows: 


Let W = nodoad watts input, 

R = effective armature resistance, 

E = no-load direct e.m.f., 

I = any chosen value of load current; 
then I 2 R = copper loss of armature due to load, 
E — I R — apparent external direct e.m.f., 

W + E I = input, 

El — PR = output, 

El-PR 

-ettw - etficiency ’ 


which becomes a maximum when PR = W, as a close approx¬ 
imation. 

'It should be noted that the losses are IV + PR, and that while 
the ratio of E 1 to W + PR is a maximum when PR = W, at 
any armature load current, 7, the input is IE + W and not 
simply I E. 

The above equations are based on the assumption of constant 
iron, friction and windage loss, which assumption is closely 
exact as stated above. In addition to these losses, the value, W 
includes the armature copper loss for the v no-load current, and 
the field copper loss for exciting current. Since for efficient 
service the exciting current should have a constant value, it 
follows that the loss from this source decreases as the machine 
is loaded due to the fact that the direct voltage decreases, 
requiring less loss in the regulating rheostat. The no-load 
armature current is of totally an alternating nature and traverses 
the whole armature winding, and during a portion of its route 
through the armature is superposed upon that part of the 
alternating supply current which is about to be converted to 
direct current. While its effect alone upon the armature re¬ 
sistance would give a constant value of loss, when the two 
currents intermingle their combined loss is greater than the 
sum of the losses of the two considered separately, since in any 
case (x-Vy) 2 is greater than x 2 + y 2 . It is thus seen that, among 



174 


ALTERNATING CURRENT MOTORS . 


the losses which have a practically constant value, one in¬ 
creases with the load while another decreases, tending somewhat 
to keep the total at a constant value. 

From the above facts and equations it appears that the curves 
of constant losses, variable losses, output, input and efficiency may be 
constructed from the two value, no-load input and effective arma¬ 
ture resistance. 

Fig. 87 gives graphically the results of calculations of the 
characteristics of a certain rotary converter of which the no-load 
losses are 1000 watts and effective armature resistance .125 ohm. 

Six-Phase Converters. 

Due to the fact that the alternating current of the motor 
portion of the converter flows in general in a direction opposed 
to that of the direct-current generator portion, the effective 
armature resistance for polyphase converters is less than that 
of the same machine used as a direct-current generator. The 
ratio of effective armature resistance to its true generator value 
is as follows: 


2 rings converter, 

1.39 

3 “ 

.56 

4 “ “ 

.37 

6 “ 

.26 

8 “ 

.21 


The ratio of effective to true armature resistance depends 
upon the number of phases. If R a represents the true arma¬ 
ture resistance, and R the effective armature resistance, and we 
assume the full-load rating of the machine to be governed 
wholly by the heating of the armature conductors, then the 
output current will be greater as a rotary converter than as a 
generator by the ratio, 

1 to 

The values of ^ and yj ~ are as follows: 

Ra 
R 

Three-phase rotaries.1.80 

Quarter-phase rotaries.2.66 

Six-phase rotaries.3.76 


\l 


R a 


R 
1.34 
1.63 
1.94 














MOTORS AND CONVERTERS. 


175 


The above facts bring forward another of equal importance. 
It is evident from the figures just given that an armature of a 
converter connected up six-phase will give a much larger output 
than when used three-phase. The theoretical ratio is about 
1 to 1.45. In practice this would be slightly modified by 
wattless currents, if such be present. 

The first thought of the use of six-phase converters suggests 
numerous complications of connections, which, upon further 
investigation, are found not to exist. With reference to the 
converter proper, the only change necessary is the addition of 
three more collector rings at a very small expense. 

An examination of the connections of a six-phase armature will 
reveal the fact that, if only alternate rings be considered, ne¬ 
glecting for the moment the additional three rings, we have 
a true three-phase armature. Now considering only the other 
three rings alone, we have again a true three-phase armature. 
Further examination will show that at any given instant the 
e.m.f. between two rings of one set chosen as above, is in direct 
phase opposition to the e.m.f. between the corresponding two 
rings of the other set; this is true of each pair of rings of each set. 
We therefore find that the three-phase e.m.f. in one set is dis¬ 
placed just 180 degrees from the three-phase e.m.f. in the other 
set. The connections to obtain six-phase currents from the two 
independent three-phase circuits are obvious from this explana¬ 
tion. 

Six-Phase Transformation. 

The flexibility of polyphase circuits in general is well exem¬ 
plified by the numerous interconnections of transformer coils 
which may be employed to produce six phases from two or 
three phases. The transformation from three to six phases 
may be accomplished by the use of three transformers, each 
having one secondary connected “ star ” fashion, or by three 
transformers, each having two secondaries, connected in “ star,” 
“ delta,” or “ ring or by the use of two transformers, each 
with two secondaries, connected in “ delta ” or “ tee or two 
transformers each with one secondary may be used as com¬ 
bined autotransformers and transformers to obtain the desired 
conversion. A few of the methods of transformation just men¬ 
tioned are of interest only from an academic point of view, and 
such will not be further discussed; only those which possess 


176 


ALTERNATING CURRENT MOTORS. 


points of special interest or are of practical value will be con¬ 
sidered in detail. 

In many respects a six-phase system may be represented as 



Transformers, Tee Secondary. 


two superposed three-phase systems, and a certain degree of 
simplicity in tracing the transformation circuits may be ob¬ 
tained by keeping this fact in mind. This fact is somewhat 



Transformers, Tee Secondary. 

emphasized by the method of transforming from two to six 
phases that is shown by Fig. 88. As will be observed, the two 
transformers are wound quite similarly to those used in the 



















































































MOTORS AND CONVERTERS. 


177 


Scott method of transforming from two to three phases; the 
difference being in the division of each secondary winding into 
two parts. The six secondary coils are connected so as to form 
two three-phase systems. These two systems are twice re¬ 
versed with reference to each other; electrica ly at the trans¬ 
formers, and mechanically at the six-phase receiver, so that the 
separate tendencies to motion would be in the same direction 
of rotation. Since there exists no interconnection between the 
two three-phase circuits the cross e.m.fs. shown in Fig. 88 can 



Figs. 90-93.—Symmetrical Voltage Diagrams, 

be observed only when the six-phase receiver of itself tends tc 
produce such e.m.fs. 

As a comparison between Fig. 88 and Fig. 89 will show, a 
relatively slight change in the primary coil of one transformer 
renders the two-phase to six-phase connections of circuits ap¬ 
plicable to transformation from three to six phases. The re¬ 
sistances shown in Fig. 89 are not essential to the operation of a 
six-phase rotary converter or similar apparatus, though when 
such apparatus is absent the existence of e.m.fs. in six-phase 
relation can best be proved by the use of a voltmeter when re- 


















178 


ALTERNATING CURRENT MOTORS. 


sistance is thus used. By varying the points on the separate 
resistances which are joined together a variety of superposed 
three-phase e.m.fs. may be obtained, the existence of which 
may be proved by a voltmeter. Figs. 90 to 93 indicate a few 



Fig. 94.—Three-phase to Six-phase Transformation; Delta 

Primary, Star Secondary. 


of the symmetrical figures thus produced. Under operating 
conditions the generator action of a six-phase rotary converter 
causes the e.m.fs. to assume the relation shown by Fig. 91, 
and the use of resistance for such purpose is entirely superfluous. 



Fig. 95. —Three-phase to Six-phase Transformation; Delta 
Primary, Delta Secondary. 


Perhaps of the many methods for transforming from three to 
six phases, the one possessing the greatest simplicity in trans¬ 
former circuits is that in which the secondaries are star con¬ 
nected, and the primaries either star or delta connected. Fig. 

































































MOTORS AND CONVERTERS. 


179 


94 shows such interconnection of circuits with the primaries 
connected in delta. With the six-phase receiver absent, a volt¬ 
meter would register only three separate single-phase e.m.fs., 
and would indicate no cross e.m.fs. between the phases. By 
tapping each secondary coil at its middle point, and joining 
these three points to form a common neutral, all the e.m.fs. of 
the symmetrical six-phase receiver will be properly indicated 
on a voltmeter. As stated previously, however, the operation 
of a six-phase receiver does not depend upon the production 
of the symmetrical figure external to the receiver and the per¬ 
formance will be quite satisfactory without joining the three 
neutral points of the separate single-phase circuits. 

Fig. 95 indicates connecting circuits for three-phase to six- 



Fig. 96. —Three-phase to Six-phase Transformation; Delta 

Primary, Ring Secondary. 


phase transtormation, both primary and secondary coils being 
connected in delta. No change whatever need be made in the 
connections of the secondary circuits in order to operate the 
primary coils in star, though the e.m.f. per primary coil would 
thereby need to be decreased in the ratio of V3 to 1, of course. 

A comparison of Fig. 95 and Fig. 96 will reveal the fact that 
the same transformers may be used for either delta or ring 
connected secondaries, though a change in the ratio of primary 
to secondary turns per coil would be necessary in order to 
operate the receiver at the same e.m.f. for the two methods of 
transformation; the change being as the ratio of the side of an 
equilateral triangle to that of a regular hexagon inscribed 
within the same circle. 


















































ISO 


ALTERNATING CURRENT MOTORS. 


Relative Advantages of Delta and Star-Connected Pri¬ 
maries. 

Since, when three transformers are connected in delta, one 
may be removed without interrupting the performance of the 
circuit—the other two transformers in a manner acting in series 
to carry the load of the missing transformers—the desire to 
obtain immunity from a shut-down due to the disabling of one 
transformer has led to the extensive use of the delta connection 
of transformers especially on the low potential six-phase side. 
It is to be noted in this connection that in case one transformer 
is crippled the other two will be subjected to greatly increased 
losses. If three delta-connected transformers be equally loaded 
until each carries 100 amperes, there will be 173 amperes in 
each external circuit wire. If one transformer be now removed 
and 173 amperes continues to be supplied to each external circuit 
ware, each of the remaining transformers must carry 173 amperes, 
since it is now in series with an external circuit. Therefore, each 
transformer must now show three times as much copper loss as 
when all three transformers were active, or the total copper loss is 
now increased to a value of six relative to its former value of three. 

A change from delta to star in the primary circuit alters the 
ratio of the transmission e.m.f. to the receiver e.m.f. from 1 to 
VS. On account of this fact, when the e.m.f. of the transmission 
circuit is so high that the successful insulation of transformer coils 
becomes of constructive and pecuniary importance, the three- 
phase line side of the transformers is frequently connected in star. 

W hen rotary converters are employed for supplying power 
to lighting circuits, it is frequently desirable that a lead at 
neutral potential be run on the direct-current side of the sys¬ 
tem. Since the input side of the converter is electrically con¬ 
nected to the output side, it follows that the neutral point on 
the alternating-current end of the converter is simultaneously 
the neutral point on the direct-current end. Por this reason, 
it is sometimes advantageous to join the low-potehtial windings 
of the transformers in such a manner as to allow an electrical 
connection to be made to the neutral point from the neutral 
conductor of the three-wire direct-current system. Thus for a 
three-ring converter the coils could be joined in “ tee ” or in 
“ star,” for a four-ring converter the coils could be intercon¬ 
nected at the central points, while for a six-ring converter the 
coils could be arranged in double interconnected “ tee ” or “ star ” 


ALTERNATING CURRENT MOTORS, 

By A. S. MCALLISTER. 

{Supplement to Chapter on Synchronous Motors and Converters). 

CIRCULAR CURRENT LOCI AND V-CURVES OF THE 

SYNCHRONOUS MOTOR. 


Although the characteristics of the synchronous motor have 
been familiar to electrical engineers even longer than have those 
of the induction motor, yet considerably more has been written 
concerning the performance of the latter machines than of the 
former. Doubtless a large portion of the difference in the atten¬ 
tions paid to these two types of machines is due to the relatively 
greater commercial importance of the induction motor, but at 
least a small part of the difference may be attributed to the 
fact that the polyphase induction motor possesses characteristics 
similar to those of a constant-potential stationary transformer 
and of a shunt-wound direct-current motor, and its performance 
can easily be explained by analogy to persons familiar with these 
two types of electrical apparatus, while the characteristics of 
the synchronous motor are essentially different from those of 
any other machine. On account of its constant-speed features 
the synchronous motor is becoming of increasing importance for 
frequency converter work, while its control of the wattless 
component of the current taken by it from the supply system 
will probably lead to its frequent use hereafter as.a “ synchron¬ 
ous condenser.” 

In view of the facts just stated, it is believed that a description 
of certain simple circular current loci of the synchronous motor 
which allow its characteristics to be determined equally as readily 
as does the circular current locus of the induction motor, will 
prove of interest at the present time. 

It is believed that the method outlined below is particularly 
advantageous in that it eliminates all unnecessary complexity. 
Moreover, the current loci being similar in many respects to the 
well-known “ circle diagram ” of the induction motor, allow the 
characteristics of this machine and those of the synchronous 
motor to be directly compared with entire simplicity. 

For the purpose of most readily developing the current loci 
used below, consider first the simple familiar case of two al- 

180a 




1806 


ALTERNATING CURRENT MOTORS. 


ternating-current generators of equal rating and exactly simi¬ 
lar in all respects. Assume these two machines to be electrically 
connected in parallel and mechanically driven by two similar and 
equal prime movers. The active voltage of each machine will 
at each instant be equal to that of the other machine; the two 
alternators will supply equal amounts of power to the external 
circuit, and there will be no cross flow of current between tne 
two machines. If the exciting current of one machine is in¬ 
creased while that of the other is unchanged and no change is 
made in the adjustment of the driving engines, then a certain 
amount of cross-current will exist between the machines, but 
they will continue to receive equal amounts of power from the 
prime mover and to deliver practically equal amounts of powei 
to the external circuit. That is to say, the power component 
of the current of each machine will be practically equal to that 
of the other; there will exist, however, a certain component of 
wattless current which traverses only the local circuits including 
the two generators. The latter current lags behind the e.m.f. 
of the over-excited generator and leads the e.m.f. of the other 
generator. Thus it tends to demagnetize the field of the former 
generator and to increase the field magnetism of the latter. 
Stable conditions are reached when the decrease in the generated 
e.m.f. of the former, and the increase in the generated e.m.f. 
of the latter alternator are such that the difference be¬ 
tween the two is just sufficient to force through the local im¬ 
pedance of the two armatures and their inter-connecting circuits 
that amount of current required to produce the necessary change 
in the field strength of the two machines. It is seen, therefore, 
that the value of the cross-current for a certain change in exciting 
current depends upon the ratio of the number of turns in the 
field coils to the number of turns on the armatures, and upon 


the local impedance of the armature circuits; that is to say, it 
depends upon the “ armature reaction,” armature resistance 
and local magnetic reactance of the armature. The “ armature 
reaction ” refers exclusively to the effect of the armature current 
upon the field magnetism. The change in the generated e.m.f. 
is roughly proportional to the cross-current, so that the armature 
reaction may, with a fair degree of accuracy, be expressed in 
ohms as the quotient of the change in the generated volts divided 
by.the cross-amperes. Although the results obtained are not 


MOTORS AND CONVERTERS. 


180 c 


strictly in accord with facts, it is customary to consider that 
the armature reaction in ohms can be treated as an addition to 
the ohms of “ local magnetic reactance ” of the armature circuit, 
the sum of the two t being designated as the “ synchronous react¬ 
ance ” of the armature circuit. The quadrature vector sum of 
the synchronous reactance and the resistance of the armature 
is known as the “ synchronous impedance ” of the armature cir¬ 
cuit, designated herein as Z m . It should be carefully noted 
that the synchronous impedance is a fictitaus, composite quantity. 
It has no real existence; of its three components, only one, 
namely the armature resistance, is constant; both the local mag¬ 
netic reactance and the armature reaction depend upon the 
electrical space position of the armature at the instant when 
the armature current reaches its maximum. 

Referring again to the two similar alternators in parallel, 
assume that, with the field strengths of the two alternators 
adjusted to equality, the supply of steam to one engine is grad¬ 
ually decreased. The alternator driven by this engine will tend 
to decrease its speed, but it continues to operate at the same 
number of revolutions per minute as the other alternator. 
What actually does occur is that it lags behind the other alterna¬ 
tor in electrical space position, such that its generated e.m.f. 
is out of phase in electrical time-degrees from the generated 
e.m.f. of the other alternator such that the vector difference be¬ 
tween them forces through the “ synchronous impedance ” 
of the two armatures and their interconnecting circuits an amount 
of current such that its vector product with the e.m.f. of each 
alternator represents the power transferred to or from this 
alternator from or to the other machine. It will be noted, 
therefore, that the gradual conversion of an alternator from a 
synchronous generator to a synchronous motor, electrically 
considered, is accompanied by a mere change in the electrical 
time-phase position of its e.m.f. with respect to the e.m.f. of 
the svstem to which it is connected. 

The vector representation of the phenomena of synchronous 
motors is rendered extremely simple when such representation 
is based on the well known facts discussed above. In Fig. 1, 0 
G is the e.m.f. of the supply system, E g ; 0 M is the e.m.f. 
of the synchronous motor E m \ G M is the “ resultant ’ e.m.f. 
E z which produces the current M I in the synchronous im- 


180 c? 


ALTERNATING CURRENT MOTORS. 


pedance of the motor circuits Z w . The angle G M 7, 0 Z , is that 
angle whose cosine is equal to the quotient of the armature re¬ 
sistance divided by the synchronous impedance of the motor; 
for simplicity this angle will hereafter be considered constant. 
The vector product of O G and 1 M is the electrical powei re¬ 
ceived from the supply system while the vector product of O M 
and IM is the mechanical power delivered to the motor 
shaft, including magnetic and frictional losses; the difference 
between these two (equal numerically to the vectoi product of 



G 


o 


Fig. 1 and 2.—Vector diagram of current and e.m.f’s 
of synchronous motor, and circular loci of armature 
current and motor counter e.m.f. 


G M and I M ) is the power absorbed thermally in the armature 
resistance. The value of I M depends solely upon the value of 
G M: O G varies directly with the e.m.f. of the supply system, 
while O M depends solely upon the field strength of the motor. 

Assuming a certain constant value for 0 G and assigning a 
value to O M it will be noted that the locus of the point M as 
the load is varied is the arc of a circle whose center is at the point 
O. For convenience the vector of the current may be plotted 
from the point G, as shown in Fig. 2; this construction is par¬ 
ticularly advantageous in that it permits of the direct represen- 




MOTORS AND CONVERTERS. 


180 c 


tation of the time-phase relation of the two quantities that are 
most easily measurable, namely, the e.m.f. of the supply system 
and the armature current. The locus of the point I as the load 
is varied is the arc of a circle whose center is on a line between 
which and the line 0 G (prolonged) three is an angle 6 Z (whose 
cosine is equal to the quotient of the armature resistance by the 
synchronous impedance). The exact location of the center of 
the circular arc, for a certain definite synchronous impedance 
depends solely upon the e.m.f. of the supply system. That is 
to say, the value G C is found by dividing the e.m.f. of the 
supply Eg by the synchronous impedence of the motor Z w . 
The radius of the circle is determined solely by the field 
strength of the motor, or more properly, by the internal counter 
generated e.m.f. of the motor E m . Thus the length C H is 
equal to the motor e.m.f. E m divided by the synchronous im¬ 
pedance of the motor, Z m . 

Since C H is proportional to O M, it will be noted that the 
diagram may be simplified and rendered more convenient with¬ 
out loss of accuracy by omitting O M entirely and allowing C H 
to represent its relative value (but not its time-phase position). 
Application of the above considerations leads to the simplified 
diagram of Fig. 3, which is the complete operating current and 
e.m.f. diagram of a synchronous motor. 

In the diagram of Fig. 3, O G is the e.m.f. of the supply, and 
01 is the current taken by the motor, in both its true value and 
time-phase position with reference to the supply e.m.f. The 
distance O C is equal'(in amperes) to the value obtained by di¬ 
viding the supply e.m.f. E g by the synchronous impedance of 
the motor Z m . The angle GOG ( = 0 Z ) is such that its cosine 
is equal to the quotient of the resistance of the armature divided 
by the synchronous impedance. It will be seen therefore that 
the line OC represents both in value and time-phase position 
the current taken by the armature when subjected to the full 
supply e.m.f., but without any counter e.m.f. The line CH has 
the same significance as in Fig. 2, being the value (in amperes) 
obtained by dividing the counter e.m.f. of the motor E m by 
the synchronous impedance Z m . When the motor e.m.f. E m 
is equal to the supply e.m.f. E g , CH becomes equal to 0 C ; under 
any condition of excitation C H bears to 0 C the ratio of the 
motor e.m.f. E m to the supply e.m.f., E g - 


0/ ALTERNATING CURRENT MOTORS. 

Referring now to any point I on the heavy circular arc of Fig. 3. 
0 1 is the input current to the motor. 

OP is the power component of the current 


0 ~Y~ is the power factor, == cos 6 

OP XO G = I m cos 6 Eg = input watts 
I m E g cos 0-losses = output watts 

OutpUt WattS rr . 

- T - = efficiency. 

Input watts 



Fig. 3.—Circular current loci for various motor 
excitations. 


O C = “ short circuit ” current (at full speed, wihout excita¬ 
tion) 

°Ji __ mQ tp£ _ percentage excitation 

O C supply e.m.f. ^ 

O H = minimum possible armature current. 

In Fig. 3 the heavy circular arc shows a single current locus 
for a definite field excitation of the motor. Referring to Fig. 1, 
It will be recalled that the radius of the circular locus of the point 
M of the motor e.m.f. vector depends solely upon the field 
.excitation of the motor. Hence the radius of the circular locus 

















MOTORS AND CONVERTERS. 


180 g 


of the point / of the current vector in Figs. 2 and 3 likewise 
depends solely upon the field excitation of the motor. Thus 
for each value of field excitation there is a definite circular current 
locus, the center of which remains always at the point C (in Fig. 
2 or Fig. 3). The current locus passes through the point O 
when the field excitation of the motor is such that the counter 
e.m.f. E m is equal to the e.m.f. of the supply E ff - For con¬ 
venience this value of motor field excitation may be designated 
as 100 per cent., and other values may be compared therewith 
on the percentage basis. Thus for the current locus represented 
by the heavy circular arc in Fig. 3 the motor excitation is 80 per 
cent. (H C being equal to .80 O C ). Other circular current loci 
for various excitations are shown by broken lines. 

It is to be noted especially that the above discussion of the 
circular current loci of Fig. 3 relates exclusively to the input 
to the synchronous motor. For any chosen value of input the 
corresponding output can be obtained by calculation when the 
losses are known. The problem of determining the friction and 
the hysteresis and eddy current losses of the armature and the 
field circuit copper loss can be solved only when accurate in¬ 
formation is obtainable concerning the constuction of the ma¬ 
chine and the exact conditions under which it is operated. The 
determination of the copper loss of the armature is, however, 
a comparatively simple matter. As the latter loss varies with 
the square of the current, quite independent of its time phase 
position, it is convenient to plot for each value of current the 
corresponding loss directly to the same scale and on the same 
diagram as used for plotting the input power. The scales 
chosen in Fig. 3 and the subsequent diagrams have been based on 
a constant supply e.m.f. of 2500 volts, an armature circuit resist¬ 
ance of R m = 10 ohms and a magnetic reactance of X m = 20 ohm s. 
Thus the impedance of the armature circuit Z m = VR m 2 + X m 2 
= 22.36 ohms and the short-circuit current (the length O C 
in Fig. 3) is 2500 volts 22.36 ohms = 111.8 amperes. The 

D 

angle G O C has a cosine (cos 6 Z = ——) of .4472. 

z, m 

The loss for each value of current can conveniently be found 
as follows: Referring to Fig. 4 select any value of current, 
such as O /, taken here as 80 amperes; the loss occasioned by 
this current in a resistance of 10 ohms is 10 (SO) 2 = 64,000 watts, 




180 /* 


ALTERNATING CURRENT MOTORS. 


or 64 kilowatts. From the point M (at 80 amperes) erect the 
perpendicular M N equal to 64 kilowatts (to the scale correspond¬ 
ing to the supply e.m.f. of 2500 volts). Quite independent 
of its time-phase possition a current of SO amperes causes a loss 
of 64 kilowatts in the armature circuit. At a certain definite 
phase position, such that the power component of the current 
is just equal to the value corresponding to 64 kilowatts (25.6 
amperes at 2500 volts), all of the power received by the syn¬ 
chronous motor is dissipated thermally in the resistance of the 
armature circuit; this position is found at the point P, where 
a horizontal line from N intersects the 80-ampere current arc 
I PM. Consider now a current of 140 amperes; the armature 
loss is 10 (140) 2 = 196,000 watts—plotted as 196 kilowatts at 
M' N'. The point P', at the intersection of the 140-ampere 
current arc and the horizontal line from N' shows the position 
of the extremity of the 140-ampere current vector when the 
power input is just equal to the loss in the resistance of the arma¬ 
ture circuit. A sufficient number of points having been located 
by the method used with points P and P' and a curve being 
drawn through these points, there is obtained the current locus 
O P P' for “ zero mechanical power ”—meaning that value of 
input power that is just equal to the power dissipated thermally 
in the resistance of the armature circuit. From the method 
employed in its location, it may be shown that this locus is a 
true circle which passes through the point C —the “ zero excita¬ 
tion ” point—and has its center on the e.m.f. vector O G. It will 
be noted therefore that the “locus for zero mechanical power” 
is known immediately when the point C is located. It is 
interesting in this connection to note that the diameter of the 
“zero mechanical power locus” expressed in amperes (the maxi¬ 
mum current which the machine can possibly obtain from the 
supply system and just overcome its own armature copper loss) 
is equal to the quotient of the supply e.m.f. divided by the re¬ 
sistance of the armature circuit; it is greater than the “ short- 
circuit ” current at zero motor excitation in the ratio of the syn¬ 
chronous impedance to the armature resistance. These state¬ 
ments need no proof, for they will be appreciated at once from 
a study of the method used in constructing Fig. 4. 

By the use of the “ current locus for zero mechanical power ” one 
can readily determine the effective mechanical power delivered 


MOTORS AND CONVERTERS. 


180 * 


to the shaft of the synchronous motor. In Fig. 4 assume that 
with an armature current of 80 amperes the power input is 164 
kilowatts; the armature copper loss is 64 kilowatts (M N), 
hence the mechanical power at the shaft is 100 kilowatts (R P). 
By the method outlined above the point I of the “ current locus 
for 100 kilowatts mechanical power ” is located at the intersec¬ 
tion of the horizontal line R I with the 80-ampere circular arc, 
M P I. With an armature current of 140 amperes, the input 
must be 296 kilowatts to supply mechanical power of 100 kilo¬ 
watts. A second point P on the “ current locus for 100 kilo- 



Fig. 4—Circular current loci for various mechanical loads. 


watts mechanical power ” is found at the intersection of the 
horizontal line R' P (corresponding to 296 kilowatts input) 
and the 140-ampere circular arc M'P'P. A curve drawn 
through points located as have been / and P gives the complete 
“ current locus for 100 kilowatts of mechanical power deliv¬ 
ered to the armature shaft—including all losses except that of 
the armature copper. It may be shown from the method used 
in its construction that this locus is a true circle concentric with 
the circular “ current locus for zero mechanical power.” There¬ 
fore the locus for any possible 'value of mechanical power is 

















ALTERNATING CURRENT MOTORS. 


180 / 

known at once when one point, such as 7, is located on its cir¬ 
cumference. The locus for the maximum mechanical power 
which the machine can deliver to its own shaft is a circle, con¬ 
tracted to a point, at Q. This power is delivered at a power 
factor of 100 per cent and an electrical .efficiency of 50 per cent; 
the current corresponding thereto is equgl to the quotient of 
the supply e.m.f. divided by t.wice the resistance of the arma¬ 
ture circuit. These facts are well illustrated in Fig. 4. 

A comparison of Fig. 4 and Fig. 3 will show that both when 
the excitation is left constant and the load is changed, and when 
the load is left constant and the excitation is changed, the locus 
of the armature current is a true circle; in the former case the 
circle has its center at the point of zero excitation, while in the 
latter the center is at the point of maximum load. By finding 
the intersections of various constant-input circles with certain 
constant-load circles, one may readily determine the ordinates 
and abscissae for the familar so-called “ V-curves,” showing the 
relation between the armature current and the excitation of a 
synchronous motor at various loads. Such a set of V-curves * 
and a convenient method for determining the points on the 
curves are shown in Fig. 5. The ordinate 77 I of the V-curve 
for a load of 100 kilowatts at an excitation of 3500 volts is equal 
in length to the vector O 7, whose extremity lies at the intersec¬ 
tion of the 100-kilowatt current locus with the 140-per cent, 
excitation current locus; there are two points of intersection of 
these two loci, so that there are two abscissae for each ordinate 
on the V-curves. Moreover, there are two ordinates for each 
abscissa, both the V-curves and the current loci being closed 
curves. The ordinate H' I' at an excitation of 2500 volts is 
equal in length to the vector O 7' of the 100-per cent excitation 
current locus. The construction of the complete set of V-curves 
should be obvious from the above brief reference to Fig. 5. It 
is to be noted that the V-curves are plotted in an unusual posi¬ 
tion ; the diagram should be inverted for comparison with the 
more usual V-curves. 

In the above discussion there have been used certain simpli¬ 
fying assumptions that do not currespond accurately to facts 
in nature. Thus the “synchronous reactance" has been consid¬ 
ered as constant while in reality it varies throughout a consider¬ 
able range; moreover, the change in the permeability of the 


MOTORS AND CONVERTERS. 


180 k 

magnetic circuit of the motor has been neglected. The results 
obtained by the present method are identical in every respect 



to those obtained by the more usual mathematical treatment 
which are based on the same simplified assumption—within 
the limits of the errors in measuring lengths on a graphical dia- 















180 / 


ALTERNATING CURRENT MOTORS. 


gram; the latter errors are much smaller than those attributable 
to the incorrect initial assumptions. So far as reliable results 
are concerned the graphical method is equally as good as the 
analytical, while it possesses the advantage of allowing the reader 
to follow the solution step by step without losing sight of the 
involved electro-magnetic phenomena. The circular current 
loci of Fig. 5 in themselves contain all of the information imparted 
by the V-curves, and in addition thereto they show the time- 
phase relation of the supply voltage and the armature current, 
and they indicate the maximum and minimum limits and the 
critical points to much better advantage than do the V-curves. 

The treatment outlined above has been based on single-phase 
work and single-phase apparatus. It is almost unnecessary to 
call attention to the fact that the same treatment without any 
modification whatsoever is directly applicable to the operation 
of polyphase apparatus, provided equivalent single-phase values 
are used for the current, resistance, synchronous reactance and 
synchronous impedance of the armature and supply circuits. 


CHAPTER XI. 

ELECTROMAGNETIC TORQUE. 

Commutator Motors. 

% • 

Before investigating the characteristics of single-phase com¬ 
mutator motors, it is well to review a few facts relating to the 
production of torque by electromagnetic action, and to ascer¬ 
tain some method by which rotative torque can be measured 
most conveniently. 

If a bipolar, direct-current armature be placed within core 
material having uniform magnetic reluctance around the air- 
gap, as for example within an induction motor stator, and 
brushes placed upon the commutator in mechanical quad¬ 
rature, as shown in Fig. 97, be caused to carry direct current 
from two isolated sources of supply, it will be found that the 
armature has no tendency to motion in either direction, what¬ 
soever may be the values of the two currents Upon super¬ 
ficial examination one is inclined to attribute the lack of torque 
to the fact that such flux as may be in mechanical position to 
give force by its product with any current existing in the arma¬ 
ture is caused by currents in the same armature, and the two 
currents, being in the same mechanical structure, could not 
cause motion wdth reference to any external body. It will be 
evident that each current is in position to produce force in a 
certain direction due to the presence of the flux caused by the 
other current. It is not immediately apparent, however, that 
each component force thus produced tends to give motion to the 
armature with reference to the stator, and that the cause for 
the lack of resultant torque is the opposition in direction, with 
equality in value, of the component forces. 

Equality of Torques for Uniform Reluctance. 

From the fundamental law of physics that a force of one 
dyne is exerted upon each centimeter of length of a conductor 
per unit current per line cf force flux density in the area through 

181 


182 


ALTERNATING CURRENT MOTORS. 


which the conductor passes, is obtained the torque equation 
for the current through brushes /I A (Fig. 97), due to the pres¬ 
ence of the flux caused by the current through the brushes B B, 


T a =KI a $ b 

where K is a proportionality constant depending for its value 
upon the number and arrangement of the armature conductors. 
Similarly, the torque for the current through the brushes B B 


will be 


T b = K T b <j> at 


the constant, K, having the same value as above. 


N 


r 


Fig. 97.—Superposed Direct-currents in Armature; Two 
Fields in Mechanical Quadrature; No Resultant Torque. 



A study of the circuits and magnets of Fig. 97 will show that 
these two torques are opposite in direction, so that the resultant 
torque is 

T = T a — T b = K (I a <£>b — h <l>a)- 

With uniform reluctance in all directions across the air-gap 
and through the core material, the flux per unit current will 
be the same in both axial brush lines, so that 

p =p0Tl a <f> b = h<P a , 

la B 

from which is obtained T = 0. 





































ELECTROMAGNETIC TORQUE . 


1S3 


Hence, under the conditions assumed, the resultant torque 
has zero value, though each component torque may have a 
certain definite value tending to give motion to the armature. 

Inequality of Torques for Non-Uniform Reluctance. 

If the reluctance be greater in the axial line of one set of 
brushes than in that of the other, then the proportionality be¬ 
tween the current and the flux produced thereby becomes 
altered, so that I a cfib no longer equals h (fia , and the resultant 
torque assumes a value proportional to their difference. A 



Fig. 9 g.—Superposed Direct-currents in Armature; One 
Quadrature Field Neutralized; Good Operating Torque. 


change in the relative reluctance in the two directions may be 
obtained by removing a portion of the core material in one 
axial brush line, thereby retaining the projecting poles common 
in direct-current practice. 

Fig. 98 shows a method by which the flux, which current 
through the brushes A A would tend to produce, may be ren¬ 
dered of zero value for any amount of current in the circuit, 
thus giving the effect of infinite reluctance in this axial brush line. 
When the effective turns on the stator core are equal in number 
to those on the armature, with circuits connected as here indi- 





















































184 


ALTERNATING CURRENT MOTORS 


cated, the machine will operate as a separately excited direct- 
current motor. The torque will be of a value determined 
wholly by the product of the current through A A and the 
flux along B B, and will be in no wise influenced by the fact 
that the flux is produced by current in the armature. The 
statement here made easily admits of experimental verification. 

While the facts presented above are more of theoretical in¬ 
terest than of practical importance w T ith reference to direct- 
current machinery, they form the essential groundwork upon 
which are based the fundamental equations for determining 
the characteristics of numerous types of alternating-current 
commutator motors, now being developed. 

Determination of Torque by Calculation of the Output. 

A little experience with the well-known mechanical and elec¬ 
trical methods for determining torque convinces one that the 
latter method is far preferable to the former with reference to 
ease of adjustment, flexibility of operation and reliability of 
results. For ascertaining the output from either mechanical or 
electrical motors, perhaps, the most familiar method is one which 
involves the use of a direct-current generator, of which the sum 
of the input and transmission losses is taken as the value of 
the output desired. The input to the direct-current generator 
is found as the sum of its output and its internal losses. In 
order to determine the internal losses of the generator, it is 
necessary to find the value of the individual iron, friction and 
copper losses. When the resistances of the separate circuits of 
the generator and the currents flowing there through are known, 
the copper losses may readily be calculated. The armature iron 
loss varies both with the speed and the density of magnetism. 
That the effect of any change in the latter may be eliminated, 
it is usual to operate the generator as a shunt-wound machine 
with constant field excitation and with the armature brushes 
at the mechanical neutral point, under which conditions the 
iron loss will vary at a rate but sligthly greater than the first 
power of the speed, and, where the nature of the test so dic¬ 
tates, the value may accurately be determined throughout any 
desired range of speed. 

In cases where the load generator and the driving motor are 
constructed for the same e.m.f. and capacity, the output from 


ELECTROMAGNETIC TORQUE. 


185 


the generator may be fed back into the supply line, the test 
thereby using only that amount of power necessary to overcome 
the losses of the two machines. In these latter cases the in¬ 
dividual losses of each machine are calculated as formerly and 
each is subject to the same errors as before, but the sum of the 
losses, being directly measured, is accurately determined and 
may be used as a check on the separate losses. 

The method given below combines the convenience and econ¬ 
omy of the “ loading back ” method, is subject to a less number 
of sources of errors and is applicable to all types of motors, 
either direct or alternating current, which may possess either 
the series or shunt motor characteristics, and in many cases 



Fig. 99.—Determination of Torque. 


it may with equally desirable results be applied to the testing 
of either mechanical or electrical motors. 

Measurement of Torque by the Loading Back Method. 

The circuit diagram of Fig. 99 will serve to make clear the 
method of connecting the apparatus for the test and may be 
used to explain the' theory upon which the test depends. In 
Fig. 99 the load generator is shown as a constant-potential, 
shunt-wound, direct-current machine, while the driving motor, 
as shown, is a series-wound machine, and may be of either the 
direct or alternating-current type. It is desired to find the 
torque of the series machine at various speeds. If the shunt 
machine be operated as a motor being belted to the series 






































186 


ALTERNATING CURRENT MOTORS. 


machine which is run, with circuit switch C open, at a speed 
somewhat below that at which the value of the torque is desired 
to be obtained—it will require a certain armature current, / 0 , 
at a certain impressed e.m.f., E. If now the switch, C, in the 
circuit to the series machine, be closed, the shunt machine 
will be driven at an increased speed and will require an armature 
current smaller than before—perhaps of negative value—due 
to the accelerating torque transmitted to the belt, and, if the 
e.m.f., E, and the field current of the shunt machine, remain 
constant, the value of the torque, exerted by the series machine, 
expressed in equivalent watts per revolution per minute, will be: 

(/o-/ L ) E 

S 

where I Q is amp. taken by armature of shunt machine with 
switch, C, open; 

I L is amp. taken by armature of shunt machine with switch, C y 
closed; 

and S is the “ synchronous” speed of the set, as determined by 
the relation of the e.m.f. and field strength of the shunt machine. 

The equation above expresses the value of the torque by which 
the series machine assists the shunt-wound machine, and gives 
the true value of the torque which the series machine delivers 
to its own shaft. 

The convenience of this method in comparison with one which 
uses the shunt machine as a generator will be appreciated when 
it is considered that no account need be taken of the internal 
losses or of the output of the machine and that the field current 
of the shunt machine and therewith the speed of the set may 
be adjusted to any desired value for each determination of 
torque without affecting the results. The economy of the method 
is due to the fact that the set dissipates only that amount of 
power represented by the losses of the two machines, all excess 
of power being returned through the constant potential supply 
circuit by means of the current produced by the generator 
action of the shunt machine. 

The accuracy of the method depends upon the following facts: 
A direct-current motor runs at a speed such as to generate an 
e.m.f. less than the impressed by an amount sufficient to force 
through the resistance of its armature a current of a value 



ELECTROMAGNETIC TORQUE. 


187 


such that its product with the field magnetism gives the torque 
demanded at its shaft. With constant field magnetism the 
electrical torque of a direct-current machine, operated as either 
a motor or a generator, is given by the expression: 


where I is the armature current, 

E is the impressed e.m.f., 

and 5 is that speed at which the counter e.m.f. of rotation 
of the armature windings in the field magnetism equals 
the impressed e.m.f.; that is, the “synchronous” 
speed as used above. 

If Wo = watts output (electrical), 

R = resistance of armature, 
r.p.m. = actual speed of armature, 


then D = 


Wo 


IE-PR I (E-I R) I Ec 


r.p.m. r.p.m. 

Ec is the counter e.m.f. of rotation. But 


r.p.m. r.p.m. 

Ec E 


r.p.m. S ’ 


IE 


stant field strength; hence, D = —, as given above. 


where 

for con- 


Since for a certain impressed e.m.f. 5 has a definite fixed 
value for each adjustment of field strength, with constant field 
magnetism, the internal electrical torque of the shunt machine 
varies directly with the armature current, and any change in 
the value of this current serves at once as a measure of the 
change in torque exerted by the shunt machine, quite inde¬ 
pendent of all other conditions. 


Elimination of Errors. 

It remains now to show why the change in torque of the 
shunt machine may be used to determine the torque exerted 
by the driving motor. The torque delivered to the shaft of 
the series motor is less than the internal electrical torque of the 
shunt machine by that necessary to overcome the iron and 
friction losses of the shunt machine and the transmission losses 
in the belt. Since the belt and friction losses vary directly 









188 


ALTERNATING CURRENT MOTORS. 


with the speed, it will be evident that the counter torque due 
thereto will be constant. For constant field magnetism, the 
armature hysteresis loss varies as the first power, and the eddy 
current loss as the square, of the speed. Since in comparison 
with the other loss, that due to the eddy currents is relatively 
small, the sum of the iron, friction and transmission losses varies 
at a rate inappreciably greater than the first power of the speed 
and the torque necessary to overcome these losses may, for prac¬ 
tical purposes, be taken as being independent of the slight change 
in speed. The change in the internal electrical torque of the 
shunt machine, when switch C, of Fig. 99, is closed, gives at 
once the value of the torque delivered to its shaft by the series 
motor. 

The method outlined above may be used to determine the 
torque exerted by a machine when such torque is much less than 
that necessary to drive a generator of any capacity whatsoever 
and is, therefore, especially advantageous for tests where it is 
desired to find the torque at high speeds of machines possessing 
series motor characteristics. 

In Fig. 99 is shown a series-wound driving motor, but it will 
be evident that the change in torque, as given by the variation 
of the current taken by the shunt machine, may be produced 
by any type of motor. A little consideration will show that since 
at any given speed, the torque exerted by the shunt machine 
of Fig. 99 may be adjusted throughout any desired range by 
use of the field rheostat, the method may conveniently be ap¬ 
plied to motors possessing practically constant load speed char¬ 
acteristics, such as those of the direct-current, shunt-wound 
type or of the alternating-current induction type, and that 
alternating-current synchronous motors may be similarly 
tested, if after adjustment of the load on the synchronous motor 
by means of the rheostat in the field circuit of the shunt machine 
the supply of electric power be cut off from the synchronous 
machine in order to obtain the change in torque exerted by the 
shunt machine. 


CHAPTER XII. 


SIMPLIFIED TREATMENT OF SINGLE-PHASE COMMUTATOR 

MOTORS. 

The Repulsion Motor. 

Mention has already been made of the use of a commutator 
on the revolving secondary of a single-phase motor for the 
purpose of giving to the rotor a starting torque. It seems 
desirable to treat this so-called repulsion motor more in detail 
and to explain its operation more fully. The verbal descriptive 
matter presented below will serve to give to the reader a fair 
idea of the operating characteristics of the machine, after which 
the more complete analytical study of the motor may be under¬ 
taken. Since the mathematical treatment of the other types 
of commutator motors is quite the same as that used with the 
repulsion motor, it is believed that a little familiarity in the 
performance of the repulsion motor will be of great assistance 
in becoming acquainted with the characteristics of the other 
machines. 

The repulsion motor is a transformer, the secondary core of 
which is movable with respect to the primary, and the secondary 
coil of which remains at all times short-circuited in a line in¬ 
clined at a certain angle with the primary coil. Such a machine 
is represented diagrammatically in Fig. 100. Superficially con¬ 
sidered, current which flows in the secondary (the armature) 
by way of the brushes acts upon the field produced by the 
primary current to give the armature a torque which retains 
its direction with the simultaneous reversal of the two currents. 
If the brushes were placed in line with the field poles, maximum 
current would be produced in the secondary, but-it would 
have no tendency to move because such torques as are produced 
on one side of the armature would be opposed by those produced 
on the other. Similarly, if the brushes were placed at right 
angles to the field axis no torque would be obtained, for no 
current would flow in the secondary. A further analysis will 

189 


190 


ALTERNATING CURRENT MOTORS. 


show that the torque depends directly upon the product of the 
secondary (armature) current and that part of the field magnet¬ 
ism which is in' mechanical quadrature with the radial line 
joining the secondary brushes and which is in time-phase with 
the armature current. In fact, the torque follows a law similar 
in all respects to that which holds for direct-current machines. 

Electric and Magnetic Circuits of Ideal Motor. 

For the purpose of analysis, it is convenient to divide the 
primary magnetism into two components, one in mechanical 



quadrature with the line of the brushes, to give the torque, 
and one directly in line with the brushes, which produces cur¬ 
rent in the secondary by transformer action. In commercial 
repulsion -motors the primary winding is distributed over an 
approximately uniformly slotted core without the projecting 
poles shown in Fig. 100, and the assumption is made that at any 
given angle, a, to which the brushes are shifted from the field 
line, the flux component in line with the brushes is cj) cos a, and 
that in quadrature is cj) sin a, where <£ is the total primary flux. 



































191 


SINGLE-PHASE COMMUTATOR MOTORS. 

This assumption is more or less justified by the fact that when 
no current flows in the secondary, such component values of 
fluxes give a resultant equal to the primary flux and having 
the proper mechanical position on the core. As will appear 
later, however, this assumption leads to error in assigning 
values to the two flux components. In order to eliminate all 
trouble from this source and to allow the conditions to be clearly 
presented, it is well to divide the primary winding up into two 
parts placed upon two projecting cores in mechanical quadrature 
one with the other, and to locate the brushes in line with one 
core, as shown in Fig. 101. As the simplest possible case, it 



Fig. 101. —Circuits of Ideal Repulsion Motor. 

will be assumed that the number of turns on one core is the same 
as on the other and equal to the effective turns on the armature. 
Under the conditions assumed, when no current flows in the sec¬ 
ondary, the primary field would have a resultant located 45° 
from the line joining the two brushes. 

If the secondary brushes be connected together while the 
rotor is stationary, the transformer action of the flux in A will 
cause a current to flow in the secondary, which current tends 
to reduce the flux in A and allow more current to flow in the 
primary coil and to increase the flux in B. If the transformer 












































192 


ALTERNATING CURRENT MOTORS. 


action were perfect, no flux would remain in A, while the flux 
in B would assume double value and the current in the secondary 
would equal that in the primary (the primary and secondary 
turns at A being equal). Such a condition would exist if the 
primary and secondary coils were devoid of resistance and local 
reactance (magnetic leakage). 

If the resistance and local reactance at A be considered negligi¬ 
ble, when the rotor is stationary the e.m.f. across the coil on the 
core A will be zero, while that on B will be equal to the total 
impressed e.m.f. Assume that the core material at B and through 
the armature is such that the flux produced is in time phase 
with the current in the coil and proportional at all times to 
such current; that is to say, that the reluctance of the magnetic 
path of B is constant. Let it be further assumed that the 
reluctance of the magnetic circuit of A is equal to that of B. 
It should be noted that these assumptions are equivalent in 
all respects to those which are invariably implied in a math¬ 
ematical or graphical treatment where the coefficient of self- 
induction, L, is taken as constant. 

Production of Rotor Torque. 

The production of torque at the rotor can now be investigated. 
The flux in B is in time phase with the primary current, while 
the current in the armature is in phase opposition to that in 
the coil on A. Therefore, the armature current is in time 
phase w T ith the field magnetism, and the torque (the product 
of the two) retains its sign as the two reverse together. If the 
armature be allowed to move, a certain e.m.f. will be gen¬ 
erated at the brushes due to the fact that the armature conductors 
cut the field magnetism of B. This generated e.m.f. will at each • 
instant be proportional to the product of the field magnetism 
and the speed, and will, therefore, be in time phase with the 
magnetism of B. Since the resistance and local reactance of 
the armature circuit are negligible, any unbalanced e.m.f. at 
the brushes would cause an enormous current to flow through 
the armature. Such current, however, would produce a flux 
in time phase with itself and the rate of change of the flux would 
generate an e.m.f. opposing the effective e.m.f. which causes 
current to flow, the final result being that there flows just that 
amount of current, the magnetomotive force of which produces 


SINGLE-PHASE COMMUTATOR MOTORS. 


193 


a va.ue of flux the rate of change of which through the armature 
generates an e.m.f. equal and oppostie to that due to the speed. 
Now, the flux thus produced alternates through the winding 
on A and generates therein an e.m.f. equal to that similarly 
generated at the brushes and in time phase with the brush e.m.f. 
Since the flux at B is in time phase with the e.m.f. across the 
winding on B, and the e.m.f. counter-generated in the winding 
on A is in time phase with the flux in B, the e.m.f. across the 
winding B is in time quadrature to that across the A winding. 
The vector sum of the e.m.fs. at A and at B must be equal to 
the constant line e.m.f., E. 



Graphical Diagram of Repulsion Motor. 

The facts just stated lead to the very simple graphical rep¬ 
resentation of the phenomena of an ideal repulsion motor 
shown in Fig. 102, where A C represents in value and phase 
the impressed e.m.f., E ; the line A B equals the e.m.f. across 
the winding, A, at a certain armature speed, while B C is the 
corresponding e.m.f. across B at the same speed. It will be 
noted that since at any speed the e.m.fs. A B and B C must, be 
at right angles and have a vector sum equal to E, the locus of 
the point B is a true circle and can at once be drawn when A C 







194 


ALTERNATING CURRENT MOTORS . 


is located. Since the e.m.f. of the A winding is proportional to 
the value and rate of change of the flux in the core A, and such 
flux is proportional to the current in the coil, the current in 
the winding, A, is proportional to the e.m.f., A B (Fig. 102) and 
in time quadrature to it, as shown at A D. A little considera¬ 
tion will show that the locus of D is a true circle having a diam¬ 
eter A F equal to the primary current at standstill. 

It has been stated that at starting the primary and secondary 
currents were equal in value, but that under speed conditions 
another current was produced in the secondary. The value of 
this additional component of the secondary current, which must 
be such as to give the magnetism in A (Fig. 101), may be found 
as follows: The magnetism in A is proportional to the e.m.f. 
across the winding of those poles. Hence, the current to pro¬ 
duce such magnetism must be proportional to the e.m.f. and 
in time quadrature to it; or, if in Fig. 102, B C is the e.m.f. 
just referred to, C G is the component of secondary current to 
produce the corresponding flux. The locus of G is a true circle 
with a diameter equal to A F of the primary current, as will be 
apparent from what has been stated previously. The vector 
sum of the components, C G, of the secondary current and C H, 
equal and opposite to the primary current, gives the true sec¬ 
ondary current both in value and phase position. 

It is evident that at a certain speed the e.m.fs. represented by 
the lines A B and B C in Fig. 102 will be equal in value. It is 
interesting to note what occurs at this speed. Since the e.m.fs. 
generated are equal and in time quadrature, the magnetic 
fluxes must similarly have equal values and be in time quad¬ 
rature. It will be recalled that the magnetic condition at this 
speed is similar in all respects to that found in two-phase in¬ 
duction motors; that is, there is produced a true uniform ro¬ 
tating field (if a continuous core be used) moving at “ synchron¬ 
ous speed.” At other armature speeds there is produced sim¬ 
ilarly a revolving field traveling at synchronous speed, but the 
field varies in intensity from instant to instant, giving what is 
termed an “ elliptically revolving ” field, one axis of the ellipse 
remaining in line with the brushes and having a value propor¬ 
tional to, but in time quadrature with, the e.m.f. of the winding 
of B, Fig. 101, or length A B in Fig. 102, the other axis being 
in line with the field poles, A, and proportional to the e.m.f. 
across the winding on them, B C, in Fig. 102. It is noteworthy 


SINGLE-PHASE COMMUTATOR MOTORS. 


195 


that such an elliptical field is characteristic of the magnetic 
condition found with single-phase induction motors. It should 
be noted, however, that the term “ synchronous speed ” 
refers to the change in position on the core of the maximum 
magnetic flux existing at each instant and has no direct bearing 
upon the maximum speed which the rotor may attain, which 
maximum speed, in fact, is limited by conditions other than 
those here assumed to exist. 

. Calculated Performance of Ideal Repulsion Motor. 

By the use of Fig. 102 the complete performance of an ideal 
repulsion motor may be determined if values be assigned to 
the scales chosen for the current and e.m.f. Table I records 
such calculations of the performance of a certain repulsion 
motor, of which the field reactance is 10 ohms when operated 
at a constant impressed e.m.f. of 100 volts; that is, in Fig. 102, 
A C = 100 V., A F = 10 A . The method of making computa¬ 
tions is indicated at the head of each column of the table. It 
will be observed that there have been chosen certain values 
of the e.m.fs. across the field coils on the poles B (Fig. 101) 
and the corresponding values of the e.m.fs. across the trans¬ 
former coils on the poles A have been calculated. As will be 
seen, the work of determination has been much simplified by 
assuming e.m.f. values corresponding to the product of the 
impressed e.m.f. and the sine and cosine values of angles at five- 
degree intervals, so that almost all values recorded in Table I 
have been taken directly from trigonometric tables. 

The speed is found as follows: With an equal number of turns 
in the windings on A and B of Fig. 101, when the e.m.fs. across 
these windings are equal, the speed is synchronous, and this 
speed is given the value 1. Now, at other speeds, the e.m.f. 
across the A winding will be proportional to the product of the 
speed and the flux in B\ that is, the e;m.f. in the winding, B , 
as shown previously. It will be apparent, therefore, that the 
speed is the ratio of the e.m.f. across the coils on A to that 
across the coils on B of Fig. 101, or to the ratio of the lengths 
B C to A B in Fig. 103. It will be noted that this ratio is the 
cotangent of the angle A C B arbitrarily chosen at five-degree 
intervals, so that the speed may be taken at once from trigo¬ 
nometric tables. The torque is expressed in synchronous 
watts, as the ratio of output to speed. 


TABLE I.—Calculation of Performance of Ideal Repulsion Motor. E. M. F. = E — 100 volts; Inductance of Field Coil Xf 10 ohms 


196 


ALTERNATING CURRENT MOTORS 


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SINGLE-PHASE COMMUTATOR MOTORS. 


197 


Columns 6 to 11 of Table I have been calculated on the as¬ 
sumption that the brush angle a of Fig. 100 was 40°, or what is the 
same, that the number of turns on field poles B was equal to 
that on the transformer poles, A, Fig. 101, and these calculations 
are graphically represented in Fig. 103. In Fig. 104 are given 
the characteristics of an ideal repulsion motor with the brush 
angle a of Fig. 100 having a value such that its tangent is 0.25, 
and the calculations recorded in columns 6' to 11" are based 
on this assumption. The conditions here assumed are equiva¬ 
lent to what would be obtained if the number of turns on the 
field poles, B, were 0.25 of the turns on the transformer poles, 
A, Fig. 101. It is further assumed that the coils on poles B 


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Speed o| 3 | 

Fig. 103. —Characteristics of Ideal Repulsion Motor. 


are the same in number as previously; that is, that the field 
reactance is 10 ohms as before, and that the turns on both the 
armature and transformer poles have been increased to four 
times their first value. 

The method of calculation is quite the same as before, but 
perhaps a word is needed as to the calculation of the speed and 
secondary current. At synchronous speed, the magnetism in 
the field poles, B, is equal to that in the transformer poles, A, 
as noted above. Due to the increased number of turns on the 
transformer poles, at synchronous speed, the e.m.f. across the 
coiis on A will be four times that across the coils on B, Fig. 101. 
A consideration of this fact leads to the conclusion that the speed 
is all times equal to one-fourth of the ratio of B C to A B in 














































































198 


ALTERNATING CURRENT MOTORS. 


Fig. .102; that is, the speed is equal to one-fourth of the co¬ 
tangent of the angle 6, arbitrarily assumed, and hence may 
quite readily be computed. 

The component of secondary current to counterbalance the 
primary current is equal and opposite to the primary current, 
since the armature turns are equal in number to the trans¬ 
former turns on A ; but the current to produce the magnetism 
in the transformer poles under speed conditions will at all times 
be one-fourth the value required to produce the same mag¬ 
netism in the field poles, as will be seen from column 10 7 of the 
table. 

A comparison of the curves of Fig. 103 with those of Fig. 104 



Fig. 104. —Characteristics of Ideal Repulsion Motor. 


will reveal the effect of shifting the brushes from the 45° position 
farther toward the axial line of the transformer poles. It is 
essential for good performance that the angle of brush shift 
from the transformer position be quite small, usually from 12 
to 16°, depending upon the constructive constants of the machine. 

It must be very carefully noted that the curves here given 
are for an ideal repulsion motor, all resistance, local inductance 
and short-circuiting effects having been neglected, and that 
such curves cannot be realized in practice. It is worthy of 
note, however, that upon the characteristics here shown are 
based the discussions of the properties of the repulsion motor 
which have occupied so much space in the technical papers. 




































































CHAPTER XIII. 


MOTORS OF THE REPULSION TYPE TREATED BOTH 
GRAPHICALLY AND ALGEBRAICALLY. 

Electromotive Forces Produced in an Alternating Field. 

In dealing with the phenomena connected with the operation 
of alternating current motors of the commutator type, it must 
be constantly borne in mind that the machine possesses simul¬ 
taneously the electrical characteristics of both a direct current 
motor and a stationary alternating current transformer. The 
statement just made must not be confused with a somewhat 
similar one which is applicable to polyphase induction motors, 
since only with regard to its mechanical characteristics does an 
induction motor resemble a shunt-wound direct current ma¬ 
chine, its electrical characteristics being equivalent in all re¬ 
spects to those of a stationary transformer. 

Before discussing the performance of repulsion motors, it is 
well to investigate a few of the properties common to all com¬ 
mutator type, alternating current machines. It will be recalled 
that when the current flows through the armature of a direct 
current machine, magnetism is produced by the ampere turns of 
the armature current, such magnetism tending to distort the 
flux from the field poles. In the familiar representation of the 
magnetic circuit of machines,—the two pole model,—the arma¬ 
ture magnetism is at right angles to the field magnetism, the 
armature current producing magnetic poles in line with the 
brushes. The amount of this magnetism depends directly on 
the value of the armature current and the permeability of the 
magnetic path. When alternating current is used, the change 
of the magnetism with the periodic change in the current pro¬ 
duces an alternating e.m.f. which being proportional to the rate 
of change of the magnetism will be in time-quadrature to the 
current. The armature winding thus acts in all respects sim¬ 
ilarly to an induction coil. 

It is not essential that the current to produce the alternating 

199 


200 


ALTERNATING CURRENT MOTORS. 


flux flow through the armature coils in order that the alter¬ 
nating e.m.f. be developed at the commutator. Under whatso¬ 
ever conditions the armature conductors be subject to changing 
flux a corresponding e.m.f. will be generated, in mechanical line 
with the flux and in time-quadrature to it. Referring to Fig. 105 
which represents a direct current armature situated in an alter¬ 
nating field, having two pair of brushes, one in mechanical line 
with the alternating flux and one in mechanical quadrature 
thereto. When the armature is stationary an e.m.f. will be 
generated at the brushes A and A due to the transformer action 



1 , 


Line 


E.M.F. 



JJvnamo 


Transformer 

E.M.F. 


Speed E.M.F. 


Fig. 105.- 


-Electromotive Forces Produced in an Alternating Field. 


of the flux, but no measurable e.m.f. will exist between B and 
B. As seen above, this e.m.f. is in time-quadrature with the 
held (transformer) flux and as will be seen later, its value is un¬ 
altered by any motion of the armature. At any speed of the 
armature, there will be generated at the brushes B and B an 
e.m.f. proportional to the speed and to the field magnetism and 
in time-phase with the magnetism. At a certain speed this 
“ dynamo ” e.m.f. will be equal in effective value to the “ trans¬ 
former ” e.m.f. at A and A, though it will be in time-quadrature 
to it. This critical speed will hereafter be referred to as the 





































REPULSION MOTORS. 


201 


“ synchronous ” speed, and with the two-pole model shown 
in Fig. 105 it is characterized by the fact that in whatsoever 
position on the armature a pair of brushes be placed across a 
diameter, the e.m.f. between the two brushes will be the same 
and will have a relative time-phase position corresponding to 
the mechanical position of the brushes on the commutator. 

A little consideration will show that the individual coils in 
which the maximum e.m.f. is generated by transformer action 
are situated upon the armature core under brush B or B, al¬ 
though the difference of potential between the brushes B and B 
is at all times of zero value as concerns the transformer action. 
A similar study leads to the conclusion that the e.m.f. generated 
by dynamo speed action appears as a maximum for a single coil 
wdien the coil is under brush A or A. Assuming as zero posi¬ 
tion, the place under brush A and that at synchronous speed 
the e.m.f. generated in a coil at this position is e. Then the 
e.m.f. in a coil at b will equal e also. A coil a degrees from this 
position will have generated in it a speed e.m.f. of e cos a and a 
transformer e.m.f. of e cos (a ± 90) = ± e sin a. Since these 
two component e.m.fs. are in time quadrature the resultant will 

be V = V(e cos a) 2 + (± e sin a) 2 = e and is the same for all 
values of a. The time-phase position of the resultant, however, 
will vary directly with a or with the mechanical position of the 
coil. From these facts it is seen that at synchronous speed the 
effective value of the e.m.f. generated per coil at all positions 
is the same and that there is no neutral e.m.f. position on the 
commutator. 

The Simple Repulsion Motor. 

In a repulsion motor as commercially constructed, the sec¬ 
ondary consists of a direct current armature upon the commu¬ 
tator of which brushes are placed in positions 180 electrical de¬ 
grees apart and directly short circuited upon themselves, as 
shown in the two-pole model of Fig. 106. The stationary pri¬ 
mary member consists of a ring core containing slots more or 
less uniformly spaced around the air-gap. In these slots are 
placed coils so connected that when current flows in them defi¬ 
nite magnetic poles will be produced upon the field core. The 
brushes on the commutator are given a location some 15 degrees 
from the line of polarization of the primary magnetism, or 




202 


ALTERNATING CURRENT MOTORS. 


more properly expressed, the brushes are placed about 15 de¬ 
grees from the true transformer position. That component of 
the magnetism which is in line with the brushes produces cur¬ 
rent in the secondary by transformer action, and this current 
gives a torque to the rotor due to the presence of the other com¬ 
ponent of magnetism in mechanical quadrature to the secondary 
current. 

It is possible to make certain assumptions as to the relative 
values of the magnetism in mechanical line with, and in me¬ 
chanical quadrature to the brush line and thus to derive the 



Field f 


Fig. 106.—Two-pole Model of Ideal Repulsion Motor. 


fundamental equations of the machine. It is believed, how¬ 
ever, that the facts can be more clearly presented and the treat¬ 
ment simplified without sacrifice of accuracy if the assumption 
be made that the primary coil is wound in two parts, one in me¬ 
chanical line and the other in mechanical quadrature with the 
axial brush position as shown in Fig. 106. It will be noted that 
the two fields produced by the sections of the primary coil, if 
there were no disturbing influence present, would have a result¬ 
ant position relative to the brush line depending upon the ratio 
of the strengths of the two magnetisms. The angle which the 
resultant field would assume can be represented by /? having a 























REPULSION MOTORS. 203 


value such that cotan where cj) t is the flux through trans¬ 

former coil and <f>f is flux through field coil. If n be the ratio of 
turns on the transformer poles to those on the field poles, then 
for any value of current in these coils (no secondary current) 


<t± 


n or n = cotan /? 



It is understood that in Fig. 106, the core material is consid¬ 
ered to be continuous and that in the two-pole model represented 
both field poles and both transformer poles are supposed to be 
properly wound. 

In Fig. 106, let it be assumed that the machine is stationary 
and that a certain e.m.f., E , is impressed upon the primary, cir¬ 
cuits, the secondary being on short circuit. The flux which the 
primary current tends to produce in the transformer pole pro¬ 
duces by its rate of change an e.m.f. in the secondary, and this 
e.m.f. causes opposing current to flow in the closed secondary 
circuit. If the transformer action is perfect and the trans¬ 
former coil and armature circuits are without resistance and 
local leakage reactance, then the magnetomotive force of the 
armature current equals that of the current in the transformer 
coil, and the resultant impedance effect of the two circuits is 
of zero value, so that the full primary e.m.f., E, is impressed 
upon the field coil; that is to say, with armature stationary 
E t = O, and Ef = E. 

Effect of Speed on the Stator Electromotive Forces. 

It remains now to investigate the effect of speed on the 
electromotive forces of the transformer and field coils. Assume 
a certain flux </>/ in the field coil. At soeed 5 the armature con¬ 
ductors will cut this flux and at each instant there will be gen¬ 
erated an e.m.f. therein proportional to 5 <f>f, and therefore, in 
time-phase with the flux. This e.m.f. would tend to cause cur¬ 
rent to flow in the closed armature circuit, which current would 
produce magnetism in line with the brushes, and, since the 
armature circuit has zero impedance, (assumed) the flux so pro¬ 
duced will be of a value such that its rate of change through the 
armature coils just equals the e.m.f. generated therein by speed 
action. At synchronous speed, the secondary being closed, the 


204 


ALTERNATING CURRENT MOTORS. 


flux in line with the brushes must equal that in line with the 
field poles, since the e.m.f. generated by the rate of change of 
the flux in the direction of the brushes must equal that gen¬ 
erated at the brushes due to cutting the field magnetism, and at 
a speed which has been termed synchronous these two fluxes 
are equal, as previously discussed. At this speed the tw 7 o fluxes 
are equal but they are in time-quadrature one to the other. 
At other speeds the two fluxes retain the quadrature time-phase 
position, but the ratio of the effective values of the two fluxes 
varies directly with the speed. 

Fundamental Equations of the Repulsion Motor. 

Giving to synchronous speed a value of unity, at any speed, 
S, the transformer flux may be expressed by the equation 

(j) t = S <j)f (2) 

effective values being used throughout. Letting be the max¬ 
imum values of the field flux and reckoning time in electrical de¬ 
grees from the instant when the field flux is maximum, at any 
time a, the instantaneous field flux is 

<j>f = cj) cos ci (3) 

and the transformer flux is 

(j> t = 5 <f> sin a (4) 

These are the fundamental magnetic equations of the ideal 
repulsion motor. 

If at a certain speed 5, the effective value of e.m.f. across the 
field coil be F, requiring an effective flux of </>f, then across the 
transformer coil there will be an effective e.m.f. of 

T = nSF (5) 

due to the flux S $f. Since the fluxes are in time-quadrature, 
the e.m.fs. are likewise in time quadrature, so that the impressed 
e.m.f. E must have a value such that 

E = a/F 2 + P (6) 

This is the fundamental electromotive force equation of the 
repulsion motor. 

The current which flows through the field coil is 



( 7 > 



REPULSION MOTORS. 


205 


where A* is the inductive reactance of the field coil. Equation 
(7) gives the value of the primary circuit current and is the 
fundamental primary current equation. 

The secondary armature current in general consists of two 
components, that equal in magnetomotive force and opposite in 
phase to the primary transformer current, and that necessary 
to produce the flux in line with the brushes. With a ratio of 
effective armature turns to field turns of a, the opposing trans¬ 
former current is 

If — (8) 

a 


and the current which produces the transformer poles is 



5 I 
a 


(9) 


These component currents are in time-quadrature, so that the 
resultant secondary current is 

la = V7JTW ( 10 ) 

This is the fundamental equation for the secondary current. 
Combining (8), (9) and (10) 

la =-(-V / ^+S 2 ( U ) 

Cv 


It has been seen that the e.m.f. T is in time-quadrature to the 
field circuit e.m.f., F. Now the current is in time-quadrature 
with F , and hence, is in time-phase with T. Therefore, of the 
total primary e.m.f. E, the part T is in phase with the current, 
from which fact it is seen that the power factor is 


Power, 


Torque, 


cos 0 


T_ 

E 


P = E I cos 0 = 


EFT 
X E 


= IT 



I T 
S 


ISnF 


I n F 


( 12 ) 

(13) 


5 








206 


ALTERNATING CURRENT MOTORS. 


D = InF = InXI = PnX 

(14) 

£2 = F 2 + r- = F 2 (1+ S 2 n 2 ) 

(15) 

F = —- 

\/1 + 5 2 n 2 

(16) 

r- E 

X Vl + S 2 n 2 

(17) 

EVm 2 + S 2 
aAVl+S 2 n 2 

(18) 


when n — 1, that is at ^ = 45° see (1) 

E 

I a = —~ and is constant at all speeds. 
a N 

when 5=1, that is at synchronism for any value of n. 

'E 

I a = —which is seen to be equal to the primary 
a A 

current at starting (when a = 1) 
when 5=1 the secondary current 

la = “ V n 2 +\ 

LV 

and leads the primary current by angle cotan- 1 n = /? or angle 
of brush shift. See equation (1). 

Vector Diagram of Ideal Repulsion Motor. 

The above equations can be expressed graphically by a simple 
diagram as shown in Fig. 107. The diagram is constructed as 
follows: 0 E is the constant line e.m.f. O A at rt. angles to 
0 E is the line current at starting, O B A is a semicircle, O F 
in phase opposition to 0 A is the secondary current at starting. 
ODE is a semicircle. 0 G, in phase with O A, is the sec¬ 
ondary current at infinite speed. O H G is a semicircle. It will 
be noted that the ratio 0 A to 0 G is n a A and ratio of O A to 
OF is a:n. 

Distances measured from P in the direction of T represent 
speed. 

The characteristics of the machine may be found at once from 












REPULSION MOTORS. 


207 


Fig. 10/. Assuming any speed as PS, draw OS intersecting 
the circle 0 B A at B. From point G draw line G K parallel 
to 0 S. Join O and K . 

OK is secondary current; 

0 B is primary current; 

EOS is primary angle of lag; 

B C is power component of primary current; 



B C is power (to proper scale); 

O C is torque (to proper scale); 

D OK is angle of lead of secondary current. 

At synchronous speed (5=1) cotan d = n, hence scale of 
speed can readily be located. 

O D = It, see equation (8) 

OH = If. see equation (9). 








208 


ALTERNATING CURRENT MOTORS. 


The proof of the construction of diagram of Fig. 107 is as 
follows: 


T 

cos 0 = — 

E 

E 2 = T 2 + F 2 
7 = 5 n F 

p - r ( i+ N 


E = 


7 


S~ n Vl+S 2 n 2 


n 7 S n 

COS 6 = -=r = - -- 

£ Vl+S 2 ;* 2 


Power component of primary current 


F = I cos 0 = 


S n 


X (1 + S 2 n 2 ) 


Sin ff = ■* I 1 - — — = _L. 

\ 1 + S 2 » 2 Vl+S 2 n 2 

Quadrature component of primary current 

E l 


/ q = / sin d = 


x Vl +S 2 n 2 Vl + S 2 n 2 


Eq. (11) 

Eq. (6) 
Eq. (5) 

(19) 

( 20 ) 
( 21 ) 


( 22 ) 


(23) 


la = 


X (1 +S 2 n 2 ) 


(24) 


7 = cotan 0 = -f S ” : Y Q+5 2 h 2 ) 

^<7 X 1 + 5 2 w 2 E 

cotan 6 = S n (25) 

The cotangent of the angle of lag is directly proportional to 
the speed, the proportionality constant being the ratio of trans¬ 
former to field turns. 


D = ~ 


/ E cos 0 


E 2 n 


X U+5 2 n 2 ) 


= n E L 


S 


(26) 




















REPULSION MOTORS. 


209 


Torque is proportional to quadrature component of the pri¬ 
mary current (for given e.m.f.) the proportionality constant 
being the ratio of transformer to field turns. 



= P nX 


(27) 


Torque varies as the square of the primary current and in this 
respect is independent of the speed or the e.m.f. 

A comparison of equations (2G) and (27) reveals an interest¬ 
ing property of a circle. In Fig. 107, assuming the diameter 
A 0 to be unity, 0 C at all values of angle 0 equals the square 
of OB. 

From equation (27) it is seen that the torque is at all times 
positive, even when 5 is negative. Hence the machine acts as 
generator at negative speed. For the determination of the 
generator characteristics it is necessary to construct the semi¬ 
circle omitted in each case in Fig. 107. 

It is interesting to observe that the construction of the dia¬ 
gram of Fig. 107 can be completed at once when points F, 0, G 
and A and E are located. Thus the complete performance of 
the ideal repulsion motor can be determined when E, X, n and 
a are known. In the construction for ascertaining the value of 
the secondary current, it will be seen that O A'is equal to the 
vector sum of OF and 0 H, giving the vector 0 K. From the 
properties of vector co-ordinates it will be noted that the point 
K is located on the semicircle F K G whose center lies in the 
line FOG. Therefore if G and F be located, the inner circles 
F D 0 and 0 H G need not be drawn, since the point K can be 
found as the intersection of the line drawn parallel to 0 B from 
G with the circular arc F K G. 

Corrections for Resistance and Local Leakage Reactance. 

It is to be carefully noted that the above discussion refers to 
ideal conditions which can never be realized. The circuits have 
been considered free from resistance and leakage reactance while 
all iron losses, friction, and brush short circuiting effects have 
been neglected. The resistance and leakage reactance effects 
can quite easily be taken into account, but the remaining dis¬ 
turbing influences are subject to considerable error in approxi¬ 
mating their values, due primarily to the difficulty in assigning 



210 


ALTERNATING CURRENT MOTORS. 


to iron any constant in connection with its magnetic phenomena. 
It is to be regretted that the so-called complete equations for 
expressing the characteristics of this type of machinery with 
almost no exception neglect these disturbing influences, and yet 
these same equations are given forth by the various writers as 
though they represented the true conditions of operation. 

In the ideal motor the apparent impedance is 

E 

Z = T = X v'i + S’h’ (28) 


apparent resistance is 


R = Z cos 0 = X S n 


since 



L 

E 


cos 0 = —; T = S n F ; and E = y/j' 2 + F 2 


hence 


E = 


T 


5 ^Vl +S»n»; “stf = ~ 
apparent reactance is 


5 n 


+ 5 2 n 2 


(30) 


1 

vT+l^ 

Let Rf = resistance of field coil 

R t — resistance of transformer coil 
R a = resistance of armature coil 
X a = reactance of armature coil 
Xi = reactance of transformer coil 
Xf = reactance of field coil 
then copper loss of motor circuits will be 


P(Rf+R,)+I a * R a 

(31) 

. _ E Vn 2 + S 2 
a X Vl+S 2 n 2 

(18) 


X = Z sin # = X 


since 


sin 6 = 


\ 


1 - 


S 2 n 2 


1 + S 2 n 2 



E 

X Vl +S 2 n 2 


( 17 ) 

















REPULSION MOTORS. 


211 


hence 


In = 


I V n 2 + S 2 


a 


(32) 


and copper loss will be 


[<*- + *,) + R.]-,’R 


m 


(33) 


where R m is the effective equivalent value of the motor circuit 
resistance, that is 


R m = Rf+R, + (~JOj Ra 


a 


(34) 


Similarly it may be shown that the effective equivalent value 
of the leakage reactance of the motor circuits is 


X m — ^ 


” 2+S2 ) w 


a 


(35) 


If these values be added to the apparent resistance and react¬ 
ance of the ideal motor the corresponding effects will be repre¬ 
sented in the resultant equations thus 


and 


R. = XSn+Rf+R' + O-UPj R a 


X = X + Xf + X , 




(36) 


(37) 


Z = \Zr 2 + X 2 from (36) and (37) 


(38) 


R E 

cos 0 = —; / = 

Z z 


(39) 


Input — El cos 6 

output — El cos Q — F R m = P 


(40) 

(41) 


P 


torque = — = D, etc. 


(42) 










2 12 


ALTERNATING CURRENT MOTORS. 


Brush Short-Circuiting Effect. 


It will be noted that the short circuiting by the brush of a 
coil in which an active e.m.f. is generated has thus far not been 
considered. Referring to Fig. 106, it will be seen that at any 
speed 5 there will be generated in the coil under the brush by 
dynamo speed action an e.m.f. 

E s = K 4> t S (43) 

where K is constant. This e.m.f. is in time-phase with the 
flux <j ) t . In this coil there will also be generated an e.m.f. by 
the transformer action of the field flux, such that, 


Ef = K <j)f (44) 

This e.m.f is in time-quadrature to <j>f. Since </>/ and cf> t are 
in time quadrature the component e.m.fs. acting in the coil 
under the brush are in time-phase (opposition) so that the re¬ 
sultant e.m.f. is 


E h = Ef — E s = K(<j)f — S <$>;) (45) 

E b = K <j>f (1 - S 2 ) Eq. (2) (46) 


Since for constant frequency of supply current, F is propor¬ 
tional to <j)f we may write <f>f = C F, C being a constant depend¬ 
ing on the number of field turns. 


hence 


<Pf = CF = 


E,, = 


CE 

Jiq. (16) 

(47) 

Vl+S 2 n 2 

KCE (1-S 2 ) 

V1 + S 2 n 2 


(48) 


which becomes zero at ± S = 1, that is at synchronizing when 
operated as either a motor or a generator. Above synchronism 
E b increases rapidly with increase of speed. 

The friction loss can best be taken into account by consider¬ 
ing the friction torque as constant ( = d) and subtracting this 
value from the delivered electrical torque so that the active 
mechanical torque becomes, 

Torque = D—d ( 49 ) 

While the effect of the iron loss is relatively small as concerns 
the electrical characteristics of the machine it is obviously in- 






REPULSION MOTORS. 


213 


correct to neglect it when determining the efficiency. For pur¬ 
pose of analysis it is convenient to divide the core material into 
three parts, the armature the field and the transformer portions. 
Since the frequency of the reversal of the flux in both the trans¬ 
former and the field portions is constant the losses therein will 
depend only upon the flux. Thus considering hysteresis only, 
the .ransformer iron loss is 

Ii t = L <jV- 6 (50) 

w T here L is a constant depending upon the mass of the core 
material. Similarly the field iron loss is 

H f = M <pf 1-6 (51) 

M being a constant 

H t + H f = < j>?-* (M + SL) Eq. (2) (52) 

Since both the field and the transformer fluxes pass through 
the armature core and these two fluxes are of the same frequency 
but displaced in quadrature both in mechanical position and in 
time-phase relation, the resultant is an elliptical field revolving 
always at synchronous speed, having one axis in line with the 
transformer and the other in line with the field, the values being 
y/2 "<£< an d \/2~</>f respectively. The value of the two axes may 
be written thus 

V2S 4>f an d 

At synchronous speed of the armature the two become equal 
and since no portion of the iron is then subjected to reversal 
of magnetism the iron loss of the armature core is of zero value. 
At other speeds, while the revolving elliptical- field yet travels 
synchronously, the armature does not travel at the same speed, 
so that certain sections of the armature core are subjected to 
fluctuations of magnetism while others are subjected to com¬ 
plete reversals, the sections continually being interchanged. 
It is due to this fact that no correct equation can be formed to 
represent the core loss of the armature at all speeds, since the 
behavior of iron when subjected to fluctuating magnetism cannot 
be reduced to a mathematical expression. 

Observed Performance of Repulsion Motor. 

Fig. 108 shows the observed characteristics of a certain four- 
pole repulsion motor when operated at 22\ cycles, the syn¬ 
chronous speed being 675 r.p.m. It will be noted that up to 


214 


ALTERNATING CURRENT MOTORS. 


either positive or negative synchronism the apparent reactance 
is practically constant and the resistance varies directly with 
the speed, but that beyond synchronism the reactance tends 
to increase and the resistance is no longer proportional to the 
speed, the power factor tending to decrease. The detrimental 
effects above synchronism may be attributed largely to the e.m.f. 
eenerated in the coil short circuited bv the brush, as indicated 
in equation (48). 

Compensated Repulsion Motor. 

A type of motor closely related to the repulsion machine in 
the performance of its magnetic circuits is the compensated 



repulsion motor shown in Fig. 109. Its electrical circuits seem 
to be those of a series machine with the addition of a second 
set of brushes, A A, placed in mechanical line with the field coil 
and short-circuited upon themselves. The transformer action 
of this closed circuit is such that the real power which the motor 
receives is transmitted to the armature through this set of 
brushes, while the remaining set, B B, which in the plain series 
motor receives the full electrical power of the machine, here 
serves to supply only the wattless component of the apparent 
power. This complete change in the inherent characteristics 
of the series machine by the mere addition of two brushes ren¬ 
ders the study of this type of motor especially interesting. 

























































































REPULSION MOTORS. 


215 


For purpose of analysis, assume an ideal motor without re¬ 
sistance or local leakage reactance and consider first the con¬ 
ditions when the armature is at rest. When a certain e.m.f., E, 
is impressed upon the motor terminals, the counter magnetizing 
effect of the current in the brush circuits, A A, is such that the 
e.m.f. across the transformer coil is of zero value, while that 
across the armature is E. Thus when S = 0, letting E t = trans¬ 
former e.m.f. and E a = armature e.m.f., 

E t = 0, and E a = E (53) 



Fig. 109. —Two-pole Model of Compensated Repulsion Motor. 

It is evident also that when 5 = 0 the flux through the 
armature in line with the brushes A A will be of zero value, so 
that 

<j>t = 0 (54) 

Let (f>j be the flux through the armature in line with the 
brushes B B. This flux, neglecting hysteretic effects, is in time- 
phase with the line current and produces by its rate of change 
through the armature turns a counter e.m.f. of value Ef = E, 
giving to the armature circuit a reactance when stationary, of X. 



































216 


ALTERNATING CURRENT MOTORS. 


The relation which exists between the flux, the frequency, and 
the number of armature turns can be expressed thus, 


_ 2 7z f N p m 
f v2 10* 

where 

/ = frequency in cycles per second 
N = effective number of armature turns 
p m = maximum value of flux. 


(55) 


If C be the actual number of conductors on the armature, the 

C 

actual number of turns will be — These turns are evenly dis- 

4 

tributed over the surface of the armature, so that any flux which 
passes through the armature core will generate in each individual 
turn an e.m.f. proportional to the product of the cosine of the 
angle of displacement from the position giving maximum e.m.f. 
and the value of the maximum e.m.f. generated by transformer ( 
action in the position perpendicular to the flux, or the average 

e.m.f. per turn will be £ times the maximum. The — turns are 

connected in continuous series, the e.m.f. in each half adding in 
parallel to that in the other half, so that the effective series turns 

equal Thus, finally 
4 




(56) 


and Ef = 


C pm f 
V2 10 8 


(57) 


The value of the reactance will depend inversely upon the re¬ 
luctance of the paths through which the armature current must 
force the flux. The major portion of the reluctance is found in 
the air-gap, and with continuous core material and uniform air- 
gap around the core, the reluctance will be practically constant 
in all directions and will be but slightly affected by the change 
in specific reluctance of the core material, provided magnetic 
saturation is not reached. In the following discussion it will be 
assumed that the reluctance is constant in the direction of both 






REPULSION MOTORS. 217 

sets of brushes, and that the core material on Doth the stator and 
rotor is continuous. 


Apparent Impedance of Motor Circuits. 

When dealing with shunt circuits it is convenient to analyze 
the various components of the current at constant e.m.f., or 
assuming an e.m.f. of unity, to analyze the admittance and its 
components. When series circuits are being considered, how¬ 
ever, the most logical method is to deal with the e.m.fs. for con¬ 
stant current, or to assume unit value of current and analyze 
the impedance and its various components. In accordance with 
the latter plan, it will be assumed initially that one ampere 
flows through the main motor circuits at all times and the 
various e.m.fs. (impedances) will thus be investigated. 

An inspection of Fig. 100 will show that one ampere through 
the armature circuit by way of the brushes B B will produce a 
definite value of flux independent of any changes in speed of the 
rotor, since there is no opposing magnetomotive force in any 
inductively related circuit. From this fact it follows that on 
the basis of unit line current <j) a has a constant effective value, 
although varying from instant to instant according to an 
assumed sine law. As will appear later, while both the current 
through the armature and the flux produced thereby have un¬ 
varying effective values and phase positions, the apparent re¬ 
actance of the armature is not constant, but follows a parabolic 
curve of value with reference to change in speed. 

When the armature travels at any certain speed the conductors 
cut the flux which is in line with the brushes B B and there is 
generated at the brushes A A an electromotive force proportional 
at each instant to the flux (j)f and hence in time-phase with cf>f, or 
with the armature current through B B. 

Let <j) m — maximum value of <fif, then the maximum value of 
the e.m.f. generated at A A due to dynamo speed action will be, 


F = 

m 


C (f>m v 

10 8 


CoS) 


where V is revolutions per second. The virtual value of this 
electromotive force will be 


-p _ C <f>m V 

v ~ V2 10 8 


( 59 ) 




218 


ALTERNATING CURRENT MOTORS. 


A comparison of (59) and (57) will show that at a speed V 
revolutions per second such that V = / in cycles per second, 
E v = Ef for any value of <j> m . Consequently, the speed e.m.f. 
due to any flux threading the armature turns, at synchronism 
becomes equal to the transformer e.m.f. due to the same flux 
through the same turns. Ef is in time-quadrature and E v in 
time-phase with the flux at any speed, hence, E v is in time- 
quadrature with Ef or in time-phase with the line current. 

The brushes A A remain at all times connected directly to¬ 
gether by conductor of negligible resistance so that the re¬ 
sultant e.m.f. between the brushes must remain of zero value. 
On this account when an e.m.f. E v is generated between the 
brushes by dynamo speed action, a current flows through the 
local circuit giving a magnetomotive force such that the flux 
produced thereby generates in the armature conductors by its 
rate of change, an e.m.f. equal and opposite to E v . This flux, 
(j) t , is proportional to E v and being in time-quadrature thereto, 
is in time-phase with Ef, or in time-quadrature with (f>f . 


Fundamental Equations of the Compensated Repulsion 

Motor. 

From the transformer relations it is seen that 


Cy/24>tf 
= \/2 id 8 


(60) 


where \/2 0/ is maximum value of flux due to current through 
brushes AA. See (57). 

E v = G <f>t (61) 

where G is a proportionality constant. 

Let 5 be the speed, with synchronism as unity, then 

E v = S E f (62) 

and 

<j>t = S <j)f, (63) 

effective values being used. This is the fundamental magnetic 
equation of the compensated repulsion motor. 

Flux <j> t passes through the transformer turns on the stator in 
line with the brushes A A as shown in Fig. 109 and generates 
therein by its rate of change an e.m.f. Ef such that 


E t = n E v 


( 64 ) 



REPULSION MOTORS. 


219 


where n is the ratio of effective transformer to armature turns. 
Tins e.m.f. is in phase with E v , in quadrature with Ef and hence 
is in phase opposition with the line current and produces the 
effect of apparent resistance in the main motor circuits. 
Combining (62) and (64) 

E t = Sn E f (65) 

Since Ef is the transformer e.m.f. in the armature circuits due 
to constant effective value of flux from one ampere, we may 
write 

Ef = X (66) 

where X is the stationary reactance of the armature circuit, so 
that the apparent resistance of the transformer circuit is 


R = SnX (67) 

Under speed conditions the armature conductors cut the flux 
in line with the brushes A A, and there is generated thereby an 
e.m.f. which appears as a maximum at the brushes B B. This 
e.m.f. is in phase with <j> t , in quadrature with <j>f and in phase 
opposition to Ef. If E s be the value of this e.m.f. we may 
write 



C \/'2 cfi, S 
V2 10 8 


( 68 ) 


from dynamo speed relations. Comparing (60) and (68) and 
remembering that / is unity in terms of speed, there is obtained 

E s = S E v (69) 

from (62) and (66) 

E s = S 2 E f = S 2 X (70) 

Therefore the e.m.f. across the armature at B B will be 

E a = Ef - E s = X (1 - S 2 ) (71) 

This e.m.f. is in quadrature with the line current and is in 
effect an apparent reactance, so that the apparent reactance of 
the motor circuits which is confined to the armature winding is 

X = X (1 — S 2 ) (72) 

The apparent impedance of the motor circuits at speed S is 


z = Vr> + X> - V(S n X) 2 + X 2 (1 -S 2 ) 3 (73) 





220 


ALTERNATING CURRENT MOTORS. 


This is the fundamental impedance equation of the ideal 
repulsion-series motor. 

The power factor is 


cos 0 = 


R_ 

Z 


_ SnX _ 

V(S».Y) ! +x ! (i-s 2 ) 2 


(74) 


The line current is 


T 


E 

Z 


_ E _ 

VS 2 n 2 X 2 + X 2 (1 -S 2 ) 2 


(75) 


The power is, 

P = E I cos 0 


E 2 S X n 

X 2 S 2 n 2 + X 2 (1-S 2 ) 2 


(70) 


It will be noted that both the power and the power factor re¬ 
verse when 5 is negative. Thus the machine becomes a gen¬ 
erator when driven against its torque. 

The wattless factor is, 


Sin 6 = 


X (1 — S 2 ) 

VS 2 n 2 X 2 + X*Jl ~ S 2 ) 2 


(77) 


and becomes negative when 5 is greater than 1, so that above 
synchronism when operated as either a generator or motor the 
machine draws leading wattless current from the supply system. 
At 5 = 1, Sin 6 = 0, which means that the power factor is 
unity at synchronous speed, as may be seen also from eq. (74). 


At 5 = 0, 



At S = 1, 



That is, at 


synchronism the line current is equal to the current at start 
divided by the ratio of transformer to armature turns. If 
n = the current at synchronism is of the same value as at 
start but the power factor which at start was 0 has a value of 1 
at synchronism. This interesting feature will be touched upon 
later. 










REPULSION MOTORS. 


221 


The torque is 


E 2 n X 


n = L = _ 

^ Vx 2 5 2 n 2 + A 2 (1 - S 2 ) 2 


= I 2 n X 


(78) 


and is maximum at maximum current and retains its sign when 
5 is reversed. 

When 5 = 0 the secondary current, I s , is n /, and is in phase 
opposition with the transformer current I. See Fig. 109 When 
5 = 1 

Is — V 7 n 2 1 2 + 1 2 .and at any . speed S, 



Fig. 110. —Characteristics of Ideal Compensated Repulsion Motor. 


h = Vn 2 P + S 2 1- = 1 vV + S 2 

T _ E VIpTs 2 

Vs 2 n 2 X 2 + X 2 (1 -S 2 ) 2 


(79) 

(SO) 


Vector Diagram of Compensated Repulsion Motor. 

In Fig. 110 are shown the results of calculations for a certain 
ideal compensated repulsion motor of which X = 1 and n = 2. 
It is seen that with speed as abscissa, the curve representing 
the apparent resistance of the motor circuits is a right line 














































































































































































































222 


ALTERNATING CURRENT MOTORS. 


while that for the apparent reactance is a parabola. At any 
chosen speed the quadrature sum of these two components gives 
the apparent impedance of the motor. Since the scale for rep¬ 
resenting the speed is in all respects independent of that used 
for the apparent resistance, it is possible always so to select values 
for the one scale such that a given distance from the origin may 
simultaneously represent both the resistance and the speed. 
This method of plotting the values leads to a very simple vector 
diagram for representing both the value and phase position of the 
apparent impedance at any speed, and for determining the 


Speed in Percent 



Fig. 111.—Characteristics of Ideal Compensated Repulsion Motor. 


power-factor from inspection. Thus at any speed such as is 
shown at G the distance 0 G is the apparent resistance, the dis¬ 
tance G P is the apparent reactance, 0 P is the apparent im¬ 
pedance while the angle P 0 G is the angle of lead of the pri¬ 
mary current and its cosine is the power-factor. 

Fig. Ill gives the complete performance characteristics of the 
above ideal compensated repulsion motor at various positive 
and negative speeds when operated at an impressed e.m.f. of 
100 volts. 

It will be noted that the armature e.m.f., which has a certain 
value at standstill, decreases with increase of speed, becomes 

















































































































REPULSION MOTORS. 


223 


zero at synchronism and then increases at higher speeds. The 
transformer e.m.f. is zero at starting, increases to a maximum 

at synchronism and then continually decreases'with increase of 
speed. 

Calculated Performance of Compensated Repulsion 

Motor. 

The inductive portion of the impedance is contained wholly 
by the armature circuit, while the non-inductive is confined to 
the transformer coil; thus the power-factor is zero at standstill, 
reaches unity at synchronism and then decreases due to the 
lagging component of the motor impedance (leading wattless 
current). In comparison with the ordinary compensated series 
motor whose armature e.m.f. is, for the most part, non-inductive 
and continually increases with increase of speed, and whose in¬ 
ductive field e.m.f. decreases continually with increase of speed 
and whose power-factor never reaches unity, the compensated- 
repulsion motor furnishes a most striking contrast. The machine 
resembles the repulsion motor in regard to its magnetic behavior, 
but the performance of its electric circuits differs from that of the 
repulsion motor due to the fact that the speed e.m.f. introduced 
into the armature circuit B B (Fig. 109) which has been sub¬ 
stituted for the field coil of the repulsion motor (see Fig. 106) 
is in a direction continually to decrease the apparent reactance 
of the field circuit and thus to decrease the inductive component 
of the impedance of the circuits and to improve the power factor 
and the operating characteristics. It is an interesting fact that 
under all conditions of operation the e.m.f. in the coils short 
circuited by the brushes B B is of zero value, so that no ob- 
jectional features are introduced by substituting the armature 
circuit for the field coil of the repulsion motor, while the per¬ 
formance is materially improved. Experiments show that even 
with currents of many times normal value and at the highest 
commercial frequency no indication of sparking is found at the 
brushes B B. This feature will be treated in detail later. 

An inspection of Fig. 110 and of equation (73) will reveal the 
fact that at synchronism the apparent impedance is n times its 
value at standstill. If n be made unity, the apparent impedance 
at synchronism will be equal to that at standstill, while between 
these speeds it varies inappreciably. This means that from zero 


224 


ALTERNATING CURRENT MOTORS. 


speed to synchronism the primary current varies but slightly, 
and that the torque, which is proportional to the square of the 
primary current is practically constant throughout this range of 
speed. " These facts show that a unity ratio compensated-repulsion 
motor is a constant torque machine at speeds from negative to 
positive synchronism, the relative phase position of the current 
and the e.m.f. changing so as always to cause them to give by 
their vector product the power represented by the torque at the 
various speeds. Above synchronism the torque decreases con¬ 
tinually, tending to disappear at infinite speed. 

Any desired torque-speed characteristic within limits can be 
obtained by giving to n a corresponding value, the torque at 
synchronism being equal to the starting torque divided by the 
square of the ratio of transformer to armature turns. 

In connection with the discussion of the expression for deter¬ 
mining the value of the torque it is well to mention the fact that 
the commonly accepted explanations as to the physical phe¬ 
nomena involved in the production of torque must be somewhat 
modified if actual conditions of operation known to exist are to 
be represented. Referring to Fig. 109, it will be noted that 
when the armature is stationary there exists no magnetism in 
line with the brushes A A, so that the current which enters the 
armature by way of the brushes B B could not be said to pro¬ 
duce torque by its product with magnetism in mechanical 
quadrature with it. Similarly, the flux in line with the brushes 
B B could not be said to be attracted or repelled by magnetism 
which does not exist. That the current through A A produces 
torque by its product with the magnetism due to current through 
B B would be contrary to accepted methods of reasoning, since 
both currents flow in the same structure, yet, as concerns the 
torque, the effect is quite the same as though the flux in line 
with the brushes B B were due to current in a coil located on 
the field core. (As shown in Fig. 106 for the ordinary repulsion 
motor.) These fundamental facts were discussed in the chapter 
on electromagnetic torque. See Fig. 98. 

Observed Performance of Compensated Repulsion Motor. 

The calculated impedance characteristics shown in Fig. 110 are 
based on arbitrarily assumed constants of a repulsion-series 
motor under ideal conditions. It is obviously impossible to ob- 


REPULSION MOTORS. 


225 


tain such characteristics from an actual motor, since all losses 
and minor disturbing influences have been neglected in deter¬ 
mining the various values. As a check upon the theory given 
above, the curves of Figs. 112 and 113 as obtained from tests of a 
compensated-repulsion motor, are presented herewith. It will 
be observed that the apparent resistance of the transformer coil 
varies directly with the speed and becomes negative at negative 
speed, while the apparent reactance of the armature decreases 
with increase of speed in either direction and, following approx- 

Speed in 100 ,R.P. M. 



Fig. 112. —Test of Compensated Repulsion Motor—Active 

Factors of Operation. 


imately a parabolic law, reverses and becomes negative at speeds 
slightly in excess of synchronism. A comparison of the general 
shape of the curves of Fig. 112 and Fig. 110 will show to what 
extent the assumed ideal conditions can be realized in practice, 
and it would indicate that, as concerns the active iactois of opera¬ 
tion, the equations given represent the facts involved. The 
neglect of the local resistance of the transformer circuit leads to 
the discrepancy between the theoretical and obsei\ed cur\es as 
found at zero speed, the latter curve indicating a certain appaicnt 
resistance when the armature is stationary. Similarly at s^n- 


siniTO 







































































226 


ALTERNATING CURRENT MOTORS. 


chronous speed the observed apparent reactance of the arma¬ 
ture is not of zero value due to the local leakage reactance oi 
the circuit. 

In the determination of the theoretical curves only active 
factors have been considered, and it has been shown that the 
apparent reactance of the motor circuits is confined to the arma¬ 
ture, while the e.m.f. counter generated in the transformer coil 
gives the effect of apparent resistance located exclusively within 
this coil. The neglected disturbing factors, the apparent re¬ 
sistance of the armature and the apparent reactance of the 
transformer, are of relatively small and practically constant 


Speed in 100 R.P.M. 

-18 -16 -14 .12 -10 -8 -6 -4 -2 0 +2 t< U 48 4]Q 412 4 14 416 418 


•10 -J 

o -i 






















a 













// 






2 

S/S 












/ 

ZsL 







"<u p' 











/ 

O 
















— Jl 

// 








-j^pp a 

•ent I 

Lsaistan 

ce 

u ® r T!' 

Armature 












-er'y 


ol 

' 1 

'liauBi 

orui-i'i 












* 

= 5 ? 



~X 



c n 





r 






er 








1 

UQ 



a 

ao 

11 


















o 

a 

-» 












































Fig. 113. —Test of Compensated Repulsion Motor—Dis¬ 
turbing Factors of Operation. 


value throughout the operating range of speed from negative to 
positive synchronism, but they become of prime importance 
when the speed exceeds this value in either dnection, as shown 
by the curves of Fig. 113 obtained from the test of a com¬ 
pensated repulsion motor giving the curves of Fig. 112. Th° 
predominating influence of the disturbing factors above svn 
chronism is attributable largely to the effect of the short circuit 
by the brushes A A (Fig. 109) of coils in which there is pro¬ 
duced an active e.m.f. by combined transformer and speed 
action. This short circuiting effect will be treated in detail 
later. 



























































REPULSION MOTORS. 


227 


Corrections for Resistance and Local Leakage Reactance. 

The resistance and local leakage reactance of the coils may 
be included in the theoretical equations as follows: 

Let R t = resistance of transformer coil 
R a = resistance of armature circuit 
R s = resistance of secondary circuit 
X t = leakage reactance of transformer 
X a = leakage reactance of armature 
X s = leakage reactance of secondary circuit 
then copper loss of motor circuits will be 


I 2 (R t +R a ) +1 s 2 R s (81) 

I s = I Vn 2 + S 2 (82) 

I 2 [R t +R a + (n 2 + S 2 )R s ] = I 2 R m (83) 

where R m is the effective equivalent value of the motor circuit 
resistance, that is, 

Rm — R t +R a + (n 2 + S 2 ) R s (84) 

Similarly it may be shown that the effective equivalent value 
of the leakage reactance of the motor circuits is 

X m = X t 4- X a 4- (n 2 + S 2 ) X s (85) 


combining equations (84) and (85) with (73) the expression for 
the apparent impedance of the motor circuits becomes 

= V(R + R m ) 2 + (X + X m ) 2 ~ 

[S u X + Rt 4- R a {vi 2 + S 2 ) R s ] 2 + 

[X (1 -S 2 )+X t +X a + (n 2 + S 2 ) X s ] 2 

i - 7 

V(SnX + R m ) 2 + [X ( 1 - S 2 ) + X m f 

- S VI X 4” R yyl 

cos 0 = -■ - -. - 

V [X (1 — S 2 ) + X m ] 2 + (S n X -\-R m ) 2 

Input = E I cos 0 

Output = EI cos 6 — I 2 R m = P 

E 2 (S nX + Rm) 

(S nX + R m ) 2 + [X (l~S 2 )+X m ] 2 


( 86 ) 

(87) 

( 88 ) 

(89) 

(90) 











228 


ALTERNATING CURRENT MOTORS. 


(S n X + R m ) 2 + [X ( 1 - S ! ) + X m f 


( 91 ) 


E 2 SnX 


= PSnX (92) 


P (S n X + R m f + [X ( 1 - S 2 ) + X m f 


torque = D = = P n X 


(93) 


The above equations, though incomplete on account of ne¬ 
glecting the brush shortening effect and the magnetic losses in 
the cores, represent quite closely the electrical characteristics 
of the compensated repulsion motor when operated between 
negative and positive synchronism, throughout which range of 
speed the disturbing factors are of secondary importance. 


Brush Short-Circuiting Effect. 


The e.m.f. in the coils short circuited by the brushes can be 
treated by a method similar to that used with the repulsion 
motor. Referring to Fig. 109, the coil under the brush A is 
subjected to the transformer effect of the flux, </>f, in line with 
the brushes, B B, and the dynamo speed effect of the flux, <j) t , 
in line with the brushes A A. 

Effective values being used throughout, the transformer e.m.f. 
will be, assuming C actual conductors on the armature, 



(94) 


in volts for one coil. See equation (57). This e.m.f. is in time- 
quadrature with <f)f . 

The dynamo speed e.m.f. in volts for one coil will be, 



(95) 


See equation (59). This e.m.f. is in time-phase with (j) t and 
hence is time-quadrature with <j>f. Thus the electromotive force 
in the coil under the brush A is 













REPULSION MOTORS. 

229 

E a = 

e ‘ es 2 10 8 ^ ^ f ~ ^ 

(96) 

But V = f S and <[> t = 5 <£f 

See equation (63), hence 

(97) 

so that 

-e- 

c* 

II 

**•* 

(98) 


£ _ 71 f /I _ C2\ 

a 2 10 8 ^ ° 

(99) 

This resultant electromotive force has a value at 
when 5 is zero, of 

standstill 


CM 

s 

II 

(100) 

See equation (66). Thus 

finally, E a = £ (1-5’) 

(101) 


When the armature is stationary the electromotive force in 
the coil short circuited by the brush A has the value given by 
equation (100), which, with any practical motor, is of sufficient 
value to cause considerable heating if the armature remains at 
rest, or to produce a fair amount of sparking as the armature 
starts in motion. At synchronous speed, however, this electro¬ 
motive force disappears entirely, and the performance of the 
machine as to commutation is perfect. As the speed exceeds 
this critical value in either the positive or negative direction, 
the electromotive force in the short-circuited coil increases rap¬ 
idly, resulting in a return in an augmented form of the sparking 
found at lower speeds and producing the disturbing factors 
shown by the curves of Fig. 113. 

Since the e.m.f. in the coil under the brush A reduces to zero 
at both positive and negative synchronism and reverses with 
reference to the time-phase position of the line current at speeds 
exceeding synchronism in either direction, it possesses at high 
speeds the same time-phase position when the machine is oper¬ 
ated as a generator as when it is used as a motor. The time- 
phase of its reactive effect upon the current which flows in the 







230 


ALTERNATING CURRENT MOTORS. 


armature through the brushes B B is of the same sign at high 
positive and negative speeds, but reversed from the phase posi¬ 
tion of the effect at speeds below synchronism. A study of the 
test curves of Fig. 113 will show the magnitude of these effects, 
and the reversal of their time-phase positions in accordance 
with the theoretical considerations. 

With reversal of direction of rotation the time-phase position 
of the flux threading the transformer coil (Fig. 109) reverses 
with reference to the line current, and hence in its reactive 
effect upon the transformer flux the current in the coil short 
circuited by the brush A becomes negative at speeds above 
negative synchronism, though positive above synchronism in 
the positive direction. At speeds below synchronism, when the 
flux is large the e.m.f. is small, and vice versa , so that the reactive 
effect is in any case relatively small and of more or less con¬ 
stant value. See Fig. 113. 

It will be noted that in analyzing the disturbing factors no ac¬ 
count has been taken of the short-circuiting effect at the brushes 
B B, Fig. 109. This treatment is in accord with the statement 
previously made that the component e.m.fs. generated in the 
coils under these brushes are at all times of values such as to 
render the resultant zero. The proof of this fact is as follows: 

The transformer e.m.f. in the coil under B due to flux, cj ) t , in 
lines with brushes A A is 


e,=Mh 

’ 2 . 10 “ 


( 102 ) 


See equation (94). This e.m.f. is in time-quadrature with <j> t . 
The dynamo speed e.m.f. is 


7T V 4>f 
2 . 10 8 


(103) 


This e.m.f. is in time-phase with cj)f, in time quadrature with 
(j> t , and is in phase opposition to ef. Thus the resultant e.m.f. is 


Eb — ef e v — ^ (/ (j)t V (j>f) (104) 


Since V = f S and <j) t = S <j>f from equations (97) and (63), 

= (105) 

E b = 0 (106) 


and 




REPULSION MOTORS. 


231 


This theoretical deduction is substantially corroborated by 
experimental evidence, as has been noted above. Even upon 
superficial examination such a result is to be expected, since the 
vector sum of all e.m.fs. in the armature in mechanical line with 
the short-circuited brushes A A must be zero, while the e.m.f. 
in the coil at brush B must equal its proper share of this e.m.f. 
or 



(107) 


A similar course of reasoning allows of the determination of 
the electromotive force under the brush A. See equation (71). 



Eg 7Z 

C 2 


71 X (1-S 2 ) 

2 C 


(108) 


for unit current. For I amperes this becomes 

Ea = w" (1-S2) (109) 

See equation (101). 

From the facts just indicated it would seem that perfect com¬ 
mutation dictates that the electromotive force across a diameter 
ninety electrical degrees from the brushes upon the armature be 
at all times of zero value. 

It has been stated that the magnetic circuits of the compen¬ 
sated repulsion-series motor are quite the same as those of the 
repulsion motor. The fluxes in line with the two brush circuits 
under all conditions are in _ time-quadrature and have relative 
values varying with the speed such that at all times 

<f>t = S <j>f (110) 

There exists, therefore, at all speeds a revolving magnetic field 
elliptical in form as to space representation. At standstill the 
ellipse becomes a straight line in the direction of the brushes B B 
(Fig. 109), at infinite speed in either direction the ellipse would 
again be a straight line in the direction of the brushes A A, while 
at either positive or negative synchronism the ellipse is a true 
circle, the instantaneous maximum value of the revolving mag¬ 
netism traveling in the direction of motion of the armature. At 
synchronous speed, therefore, the magnetic losses in the arma¬ 
ture core disappear, while the losses in the stator core are evenly 
distributed around its circumference. 




CHAPTER XIV. 


MOTORS OF THE SERIES TYPE TREATED BOTH GRAPHICALLY 

AND ALGEBRAICALLY. 


The Plain Series Motor. 

The combined transformer and motor features of commutator 
type of alternating current machinery are well exemplified in the 
plain series motor as illustrated in Fig. 114. When the rotor is 
stationary, the field and armature circuits of the motor form two 
impedances in series. Assuming initially an ideal motor without 
resistance and local leakage reactance, each impedance consists 
of pure reactance, the current in the circuit having a value such 
that its magnetomotive force when flowing through the arma¬ 
ture and field turns causes to flow through the reluctance of the 
magnetic path that value of flux the rate of change of which 
generates in the windings an electromotive force equal to the 
impressed. 

If E be the impressed e.m.f., Ef the counter transformer e.m.f. 
across the field coil and E a the counter transformer e.m.f. across 
the armature coil, when the armature is stationary 

E = Ef + E a (111) 

From fundamental transformer relations there is obtained the 
equation 

Et = ~ ' ■ see e q- ( 55 ) (112) 


where / = frequency in cycles per second 
Nf = effective number of field turns 
c pf = maximum value of field flux. 
Similarly 

2 7 :f N a <f> a 
\/ 2 10 8 



where N a — effective number of armature turns 
cj) a = maximum value of armature flux. 

232 




SERIES MOTORS. 


233 


Fundamental Equations of Series Motor with Uniform 

Air-Gap Reluctance. 

Since the field and armature circuits are electrically series con- 




Speed E.M.F. 



p IG> ii4.—Circuit and Vector Diagrams of Plain Series Motor. 

nected and are mechanically so placed as not to be inductively 
related, with uniform reluctance around the air gap the fluxes in 
mechanical line with the two circuits being due to the magneto- 































234 


ALTERNATING CURRENT MOTORS. 


motive force of the same current will be proportional to the 
effective number of turns on the two circuits. 

Therefore 


<}>l = 0a 

Nf N a 


(114) 


If n be the ratio of effective field to armature turns 

Nf = n N a (115) 

and 

4>f = n <j> a 

Let C be the actual number of conductors on the armature, 
then 

N a = (see eq. 56) (117) 

Z 7T 


Under speed conditions the armature conductors cut the field 
magnetism and there is generated by dynamo action a counter 
e.m.f. proportional to the product of the field flux and the speed 
in time-phase with the flux, in leading time quadrature with the 
field e.m.f., Ef and the armature e.m.f. E a and in phase opposi¬ 
tion with the current. 

Thus 

E v = < LtN (see eq. 59) (118) 

where V is revolutions per second of bipolar model. 

Combining (117) and (118) 


2 7T V N a fa 
V / 2”10 8 


(119) 


If S be the speed with synchronism as unity, then 

V = Sf 


and 


E v = 


2 7t S f Ngfo 

vTio 8 


( 120 ) 


( 121 ) 


combining (113), (116) and (121) 


77 E a S (j) f _ 
E v = —7 ~~ = E a S n 

(Pa 


( 122 ) 







SERIES MOTORS. 


235 


combining (112), (115) and (121) 



E f SN a 

Nt 


E f S 
n 


(123) 


comparing (122) and (123) 

Ef = n 2 E a 

Under speed conditions the impressed e.m.f. is balanced by 
three components, E v in time phase opposition with the line cur¬ 
rent and Ef and E a , both in leading time quadrature with the 
line current. 

Thus 

E = VES+{E a + Ei)> (125) 

£ = V E a 2 S 2 n- + ( E a + n- E a )- — £oVS ! k ! +(1 + h 2 ) 2 (126) 

This is the fundamental electromotive force equation of the plain 
series motor having uniform reluctance around the air-gap. 

On the basis of unit line current the electromotive forces may 
be treated as impedances, as w r as done with the repulsion-series 
motor, so that the impedance equation becomes 

Z = X a \/S 2 n 2 + (1 + n 2 ) 2 ' (127.) 


where 5 n X a = R and (1 + w 2 ) X a = X 


The power factor is 


R S n 

-- = cos 0 = . 

Z y/S 2 n 2 + (1 + w 2 ) 2 


(128) 


which reverses when 5 becomes negative and continually ap¬ 
proaches unity with increase of 5 in either direction. 

When S = 1, or at synchronism 


cos 6 = — . . - -. 

\^n 2 + (1 + w 2 ) 2 


(129) 


which when n = 1 or for unity ratio of field to armature turns 
becomes 


cos 0 = . ... — = .447, 

Vi + (i + i 2 ) 


(130) 


and decreases with either an increase or decrease of n. It is 














236 


ALTERNATING CURRENT MOTORS. 


apparent therefore that the power factor of such a machine is 
inherently very low and cannot be improved by a mere change 
in the ratio of field to armature turns. 

The line current is 


E E _-__ 

“ Z A'„ VS 2 n 2 +(l-+n 2 ) 2 


The power is 


772 

P = E I COS 0 = ~tt 


S n 

S 2 n 2 + (1 pn 2 ) 2 


(131) 


(132) 


which becomes negative when 5 reverses, or the machine oper¬ 
ates as a generator when driven against its natural tendency to 
rotation. 

The torque is 


D = Z = R 

S X 


n 


a 


C 2 2 , /I 2N2 = EnX a . (133) 
5 2 w 2 +(l + n 2 ) 2 


which is maximum at maximum current and retains its sign 
when 5 is reversed. 

At starting the torque is 


F 2 

Eq = VT 


n 


X a (1+n 2 ) 2 


At synchronous speed, the torque is 


D, = 


n 


E 2 


X a + (1+n 2 ) 


and 


Ds = (1 +n 2 ) 2 

D 0 n 2 + (1 + n 2 ) 2 


(134) 


(135) 

(136) 


which when n is negligibly small approaches a value of unity and 
when n is infinitely large also tends to reach a value of unity. 
When n = 1 equation (136) reduces to 


Ds = ( 2) 2 

D 0 1 + ( 2) 2 


(137) 


the interpretation of which is that the torque of the unity-ratio 
single-phase, plain series motor with uniform reluctance around 










SERIES MOTORS. 


237 


the air-gap varies only 20 per cent, from standstill to synchro¬ 
nism, and therefore, that such a machine is unsuited for traction. 
This statement applies to the ideal single-phase motor without 
internal losses, and must be somewhat modified to include true 
operating conditions. The method of treating the various losses 
has previously been discussed and will further be enlarged upon 
in connection with the compensated types of series machines. A 
little consideration will show that such modifications as must be 
introduced have a detrimental effect upon the characteristics of 
the machine, and tend to lay greater stress upon the statement 
just made. These facts are graphically represented in the per¬ 
formance (impedance) diagram of Fig. 114. 0 A is the power 

and A B the reactive components of the apparent field impedance 
at starting while B C and C D are the corresponding power and 
reactive components of the apparent armature impedance. The 
power component of apparent armature impedance due to 
dynamo speed action is shown asfiEorfiF, giving the resultant 
impedance under speed conditions of 0 E or 0 F and indicating 
an angle of lag of the circuit current behind the impressed e.m.f. 
of E 0 A or F 0 A. The variation in torque due to increase of 
speed from synchronism to double synchronism with a unity 
ratio constant reluctance machine, as represented in Fig. 114, 
would be as the square of the ratio of O F to 0 E. 

Fundamental Equations for Motor with Non-Uniform Re¬ 


luctance. 


An inspection of equation (136) will reveal the fact that a 
change in the value of n does not improve the torque charac¬ 
teristics of the machine unless such change be accompanied with, 
an increase in reluctance of the magnetic structure in line with 
the brushes B 1 B 2 (Fig. 114). That is to say, if the mechanical 
construction is such that equation (114) may be written 



(138) 


where m is a constant of a value many times unity, the oper¬ 
ating characteristics of the machine become much improved. 
Thus equation (116) becomes 


<j>f = ni n <j> a 


( 139 ). 


238 


ALTERNATING CURRENT MOTORS. 


and equation (122) is changed to 

E a S cj)f 


E = 


4*a 


= E a S m n — 


E f S 


n 


E a = 


Ef 


m n 


2 


E = VE v * + (E a + Ef> 




+ (TE+eA 

\m n z / 


e = e ^( v ) 2+ ( i+ ^y 

R = — Xu X= X,(l+— 2 ) 

— n — *\ m nrj 


Cos 0 = = 


R 

Z 


_s 

n 


( 140 ) 


(I4i; 


(142) 


(143) 


(144) 




(145) 


when S = 1 or at synchronism 


1 _ 

n 


Cos 0 = 


\ 


/1\ hnn 2j r 1\ 2 I ^/mw 2 +l\ 2 
\ n) \ nin 2 ) \ \ m n ) 


(146) 


With an excessively large reluctance of the magnetic structure 
in line with the brushes B x B 2 (Fig. 114), that is, with an enor¬ 
mous value of m, the power factor at synchronous speed ap¬ 
proaches 


Cos 0 = -- 

Vl +n 2 


(147) 


the interpretation of which equation is that the operating power 
factor of such a machine is largely dependent upon the ratio of 
field to armature turns. A little study will show that at any 
chosen speed, whether synchronous or not, the cotangent of the 
angle of lag is directly proportional to the ratio of armature to 
field turns, and that the power-factor, the corresponding cosine, 




























SERIES MOTORS. 


239 


can be given any desired value by a proper proportioning of the 
windings. This feature will be treated more in detail when deal¬ 
ing with compensated motors. 

The current of the high brush-line-reluctance machine is 


t _ E _ E n 
1 ~ Z = ~X f 


The power is 

P — E I cos 0 — 

The torque is 


P E 2 n 
S ~ X f ‘ 


At starting the torque is 




\ 

E 2 nS 


( mn‘+ 1 \ 2 

\ m n / 


1 


Xf ' (m n 2 + 


( m n 2 + IV 
m n ) 


P X f 


S 2 + 


1m n 2 + IV 

\ m n ) 


n 


E 2 n 


y 1m n 2 + 1\ 2 
\ m n ) 


At synchronous speed the torque is 

E 2 n 


D, 




m n 

f m n 2 + 1\ 2 


Ds = 
D n 


( mn 2 -\- 1\ 
m n ) 
1m n 2j r 1\ 

\ m n ) 


( 1 ^ 8 ). 


(149) 


(150) 


(150) 


(151) 


(152) 


1 + \ m n 

which ratio, with an enormous value of m, approaches 


D. 


nr 


(153) 


D o 1 T h 2 


the significance of which is that the change of torque from stand¬ 
still to synchronism can be altered at will by change in the ratio 
of field to armature turn and that a relatively low value of n 
would produce a machine suitable for traction. 





















240 


ALTERNATING CURRENT MOTORS. 


By using projecting field poles thus leaving large air-gaps in 
the axial brush line and thereby increasing the reluctance of the 
structure in line with the magnetomotive force of the armature 
current, the flux produced by the armature current may be ma¬ 
terially reduced, thus giving to m a relatively large value, and 


Field Core 




Fig. 115.—Circuit and Vector Diagrams of Inductively- 
Compensated Series Motor. 

the power factor will be thereby correspondingly increased with 
a resultant improvement in the torque characteristics of the 
machine. Even under the most favorable conditions, however, 
it is impossible to reduce the reactance of the armature circuit 
to an inappreciable value, that is, to give to m an enormous 


























































SERIES MOTORS. 


241 


value, due to the inevitable presence of the magnetic material 
of the projecting poles. 


Inductivei.y Compensated Series Motor. 

The most satisfactory method of reducing the inductive effect 
of the armature current is to surround the revolving armature 


Field Coil 




Fig. 116.—Circuit and Vector Diagrams of Conductively 
Compensated Series Motor. 

winding with properly disposed stationary conductors through 
which current flows equal in magnetomotive force and opposite 
in phase to the current in the armature. This compensating 
current may be produced inductively by using the stationary 














































242 


ALTERNATING CURRENT MOTORS. 


winding as the short circuited secondary of a transformer of 
which the armature is the primary, as illustrated diagrammat- 
ically in Fig. 115, or the main line current may be sent di¬ 
rectly through the compensating coil as shewn in Fig. 116. In 
the former case the transformer action is such that the com¬ 
pensation is practically complete, giving minimum combined 
reactance of the two circuits while in the latter case, the propor¬ 
tion of compensation can be varied at will. It is found that in 
any case the best general effects are produced when the com¬ 
pensation is complete, and experiments seem to indicate that 
under such conditions the two methods of compensation differ 
inappreciably for strictly alternating current work, but that 
for direct current operation where the forced compensation can 
be used to prevent field distortion and to improve the commuta¬ 
tion, the latter method is preferable. 


CONDUCTIVELY COMPENSATED SERIES MOTOR. 

Referring to Figs. 115 and 116, assume an ideal series motor 
with complete compensation, letting n be the ratio of effective 
field to armature turns; at any speed 5 with synchronism as 
unity, the apparent impedance of the motor circuits will be 


of which 



(154) 

(155) 


represents the reactance of the motor circuits which is confined 
to the field coil, and of which 


R = 


5 X f 
n 


(156) 


represents the apparent resistance effect of the dynamo speed 
e.m.f. counter generated at the brushes B x B 2 due to the cutting 
of the field flux by the armature conductors (see eq. 123). 

The power factor is 

S 


* R 

cos o = — 


n 



(157) 


which continually approaches positive or negative unity with 
increase of speed in the corresponding direction. 






SERIES MOTORS . 


243 


At synchronism when 5=1, the power factor is 


cos 0 

The line current is 


1 

V1 + n 2 


(see eq. 147) 


(158) 


I 


The power is 


E 

Z 



1 



S 


P = E I cos 0 

The torque is 


£ 2 


n 


Xf S 2 




+ 1 



PXf 

n 


(see eq. 150) 


(159) 

(160) 


(161) 


The ratio of the torque at synchronous speed to that at stand¬ 
still is 

D 1 n 2 

= (seeeq - 153) (162) 
o -f~ 1 

nr 

which in a practical machine can be made as much smaller than 
unity as desired by a proper proportioning of the field and arma¬ 
ture windings. It is evident, therefore, that such a machine can 
be made suitable for traction when a proper value of n is chosen. 

Complete Performance Equations of Compensated Motors. 

The above equations refer to ideal motors without resistance 
and local leakage reactance and devoid of all minor disturbing 
influences. A close approximation for the effect of the resist¬ 
ance and leakage reactance may be obtained as follows: 

Let rf — resistance of field coil 

r c = resistance of compensating coil (reduced to a 1 to 1, 
armature ratio) 
r a = resistance of armature 
Xf = local reactance of field coil 

x a = combined leakage reactance effect of armature and 
compensating coils 












244 


ALTERNATING CURRENT MOTORS. 


Then the apparent impedance is 
Z = 




+ rf + r c + + (Xf + xf +x a )- (163) 


Power factor is 


R 


SX f 


n 


+ rf+r c + r a 


Cos 6 = -T 7 = 


Power input is 
P= E I cos 6 = 


^ (^^ + H+r c + r°) + Xf+*f + x a 


(164) 


c 


E 2 (EL + rf + r c + r 


n 


'*) 


S Xf 


n 


+ r f +r c + »- 0 ) + (Xf + *f+*. 


•y 


(165) 


The copper loss and equivalent effective resistance loss will be 

E 2 (rf + r c + r a )_ 


7 2 X = 


( 


SXf 


W 


+ rf +r c + r^J + ^Xf + *f + x 


•y 


(166) 


Electrical output is 

P-PR ■■ 


E 2 S X 


f 


n 


( 


S Xf 


n 


+ rf +r c + r<^J + (X f + xf+ x, 


■) 


C2 (167) 


The torque is 


P-PR PXf . 

D = --- = -- (see eq. 161) 


S 


(168) 


n 


Vector Diagram of Compensated Series Motors. 

The equations here given are represented graphically in the 
diagrams of Figs. 115 and 116, which show the impedance (e.m.f. 
for unit current) characteristics of the machines. 

0 A = rf 
B C = r a + r c 
A B = Xf + xf 
DC = X a 

S Xf 

D E = -— f - at speed S 
n 

0 E = Z at speed S 

cos E 0 A =• cos 0 = power factor at speed S 




























SERIES MOTORS. 


24 5 


These characteristics together with the brush short circuiting 
effect and other minor modifying influences will be discussed 
later. It is sufficient here to state that the effect of the short 
circuit by the brush of a coil in which an active e.m.f. is gen¬ 
erated, both by transformer and speed action, tending to in¬ 
crease the apparent impedance effects at high speeds is to some 
extent balanced by the fact that the flux which causes the gen¬ 
eration of a counter e.m.f. by dynamo speed action is out of 
phase and lagging with respect to the line current and that the 
counter e.m.f. therefore, tends to lag behind the current or to 
cause the current to become leading with respect to the counter 
e.m.f., so that the neglected disturbing influences tend to render 
the final effect quite small, the result being that the incomplete 
equations and corresponding graphical diagrams as given above, 
represent quite closely the observed performance characteristics 
of the compensated series motors. 

Induction Series Motor. 

Excellent performance of the compensated alternating-current 
motor may be obtained by using the field coil as the load circuit 
from the compensating coil employed as the secondary of a trans¬ 
former, the armature being used as the primary, as diagrammat- 
ically represented in Fig. 117. The current which enters the 
armature winding through the brushes B x B 2 causes the forma¬ 
tion on the armature core of magnetic poles having the mechan¬ 
ical direction of the axial line joining the brushes, and the rate 
of change of the magnetism generates an electromotive force in 
the compensating coil. Due to this electromotive force, current 
flows through the locally-closed circuits around the compensat¬ 
ing and field coils, and produces magnetic poles in the stationary 
field-cores. 

Consider now the load-circuit surrounding the quadrature 
field-cores. Since to this winding there is no opposing secondary 
circuit, the magnetism in the core will be practically in time- 
phase with the current producing it. This current is the sec¬ 
ondary load-current of the transformer. As is true in any trans¬ 
former, there will flow in the primary coil a current in phase 
opposition to the secondary current in addition to and super¬ 
posed upon the primary no-load exciting-current. It is thus 
seen that the load-current in the primary (or armature) coil will 


240 ALTERNATING CURRENT MOTORS. 

be in time-phase opposition with the magnetism in the quadrature 
core. And, since this current and the magnetism reverse signs 
together, the torque, due to their product and relative mechan¬ 
ical position, will remain always of the same sign—though 
fluctuating in value. Hence the machine operates similarly to a 
direct-current series motor. 


Compensating 

Coil 





When the armature revolves at a certain speed, the motion 
of its conductors through the quadrature magnetic field, gener¬ 
ates in the armature winding an electromotive force which ap¬ 
pears at the brushes B x B 2 as a counter e.m.f. This weakens the 
effective electromotive force and therewith the armature-current, 
the armature-core magnetism, the field-current and the field-core 
magnetism. Thus there results from increased speed of the arma- 





















































SERIES MOTORS. 


247 


ture a reduced torque, just as occurs in direct-current series 
motors. By increasing the applied electromotive force, an in¬ 
crease of torque can be obtained even at excessively high speeds, 
and the motor tends to increase indefinitely the speed of its ar¬ 
mature as the applied electromotive force is increased, or as the 
counter torque is decreased. There is no tendency to attain a 
definite limiting speed as is found to be true with revolving 
field induction-motors and repulsion motors. 


Fundamental Equations of Induction Series Motor. 

Let E a be the counter transformer e.m.f. across the armature 
coil, the armature being stationary. 

Then 



2r,}N a (j)a 

a/ 2 10 8 


see eq. 


(55) 


(169) 


where / = frequency in cycles per second 
;V 0 = effective number of armature turns 
(j) a = maximum value of armature flux cutting the 
compensating coil 

q 

;V a = — see eq. (56) (170) 

Z 7Z 


where C is the actual number of conductors on the armature, a 
bipolar model being assumed. 


fC<!> a 

V2 10 8 


(171) 


Let N c 
then 


effective number of turns on the compensating coil, 



2nfN c <f> a 
a/ 2 10 8 


E. N c 
N a 


(172) 


where E c = transformer e.m.f. of the compensating coil. 
Let Nf = effective number of turns on the field coil 
then 


2 7 i j Nf <j)f 

V2 10 8 


( 173 ) 








248 


ALTERNATING CURRENT MOTORS * 


where Ef = impressed e.m.f. of the field coil 
c pf maximum value of field flux 

Ef = E c , hence N c <j> a = A T f (pf 

and 

±f = N c 

(pa Nf 

Let E v be the e.m.f. counter generated at the brushes B x B 2 
(Fig. 117) by speed action due to the cutting of the flux <p t by the 
armature conductors C at speed V revolutions per second, then 


t-> L & f 4^ /rn\ 

Ev = vLo 8 see eq ' (59) 

(176) 

oo 

II 

(177) 

where 5 is the speed with synchronism as unity. 
Combining (171), (176) and (177) 


r- S E a cpf S E a N c 

" (pa " Nf 

(178) 

Let n be the ratio of effective field to compensating 

coil turns. 

5 E 

E v = -- see eq. (123) 

n 

(179) 


(174) 

(175) 


This electromotive force is in time-phase with the field flux 
(pi, is in phase opposition with the line current and hence is in 
time quadrature (leading) with respect to the e.m.f. E a . The 
impressed electromotive force E is balanced by the two com¬ 
ponents, E v and E a so that 

E = VEj+E} ( 180 ) 


£ = £ “Vl + 1 (181) 

On the basis of unit line current, the electromotive forces may 
be treated as impedances, as was done with the repulsion-series 
and compensated-series motors. 



(182) 









SERIES MOTORS. 


249 


where 

— = R and X, =X 
n — — 


Xt being the combined reactance effect of the field, compensating 
coil and armature circuits. 

The power factor is, 


cos 0 = 


R 

Z 


S 


n 


S 



\/S 2 + n 2 


(183) 


w T hich when S = 1 or at synchronism, reduces to 

cos 0 = . ... 

Vl+n 2 


(184) 


the interpretation of which is that the power factor at synchro¬ 
nism can be caused to approach unity quite closely by the use of 
a small value of w, that is, by employing a small ratio of field to 
compensating coil turns. With increase of speed the power fac¬ 
tor continually increases for any value of n. 

The line current is 

EE 1 En 

^Y~X, Is 2 7 ~ x ‘\ / S r +n 2 ( 185 > 

W + 1 

The power is 

_S 

E 2 n E 2 S n 

P= EIcos()= ~ x, (S 2 + « 2 ) (186) 

9 4 ” 1 


which becomes negative when 5 reverses, or the machine oper¬ 
ates as a generator when driven against its natural tendency to 

rotation. 

The torque is 



E 2 n 

X t (S 2 + n 2 ) 


PXt 

n 


(187) 


which is maximum at maximum current and retains its sign 
when 5 is re\ ersed. 


















250 


ALTERNATING CURRENT MOTORS. 


At starting, the torque is 



E n 
X t ‘ 


( 188 ) 


at synchronous speed, the torque is 


and 



E n 
X t ’ (T+7F) 


D_ s _ n 2 
D 0 l+n 2 


which when n = 1 reduces to 


Dj _ _ 1 _ 

D 0 1 + 1 


(189) 

(190) 


(191) 


and can be given any desired value by a proper selection of n, 
see eq. (153). A relatively low value of n would produce a 
machine having the torque characteristics of the direct current 
series motor and hence one suitable for traction. See eq. (162). 

It remains to investigate the relation of the currents in the 
compensating coil and in the armature circuit (the secondary and 
primary of the assumed transformer). 

Let i a be the current w T hich would flow in the armature when 
the field coil circuit is open. Then i a is the exciting current of 
the assumed transformer and it has a value such that its product 
with the effective number of armature turns, forces the flux,# a , 
demanded by the impressed e.m.f., through the reluctance of 
their paths in the magnetic structure, in line with the brushes 
B X B 2 (Fig. 117). When the field circuit is closed there flows 
through the field and compensating coil a current if, of a value 
such that its magnetomotive force when flowing through the 
field # turns Nf, produces the flux (Dj demanded by the e.m.f. Ef or 
E c . The current if is in time-phase with the flux (Df and hence 
is in time quadrature with the e.m.f. E c . The current i a is in 
phase with the flux (D a and in time quadrature with E a or E c . 
When the field circuit is closed a current equal in magnetomotive 
force and opposite in phase to if is superposed upon i a in the 
primary (armature) circuit. These two currents are directly in 
phase so that the resultant current becomes 

I = i a + p ij 


(192) 




SERIES MOTORS. 251 

where p is a proportionality constant the value of which will be 
discussed later. 

Since both i a and if reach their maximum values simultane¬ 
ously with cpf, one is led to the highly interesting conclusion that 
even the exciting current i a is effective in producing torque by 
its direct product with thQ field magnetism, and that under 
speed conditions both i a and p if are equally effective (per 
ampere) in producing power. 

The relative values of i a and if and of p may be approximated 
as follows: 

Assuming similar conditions for the three coils, the field, the 
compensating and the armature circuits,—equal reluctance— 


i tt N a _i,Nf 

(pa <pf 

(193) 

Vi 

£ 

§ 

II 

H-*. 

(194) 

N c (p a = Nf <pf see eq. (174) 

(195) 

, Nc4> a 

N, 

(196) 

N a (pf • A^ 0 N c . A a i a 

N, 4> a la ~ Nf “ N c n 2 

(197) 


From transformer relations there is obtained the equation 

jS = p see eq. (192) (198) 


Combining (197) and (198) 

. = Jy__ 

p n 2 

Combining (199) and (192) 

z “*« ( 1+ ^) 
Comparing (199) and (200), 


' p n 2 _ / 

1 p n 2 + p 


(199) 


( 200 ) 


if = 


( 201 ) 











252 


ALTERNATING CURRENT MOTORS. 


Corrections for Resistance and Local Leakage Reactance. 

The relations above expressed depend upon certain assump¬ 
tions as to the reluctance in line with the armature circuit and 
the field coil, and will be modified if the assumptions made are 
not applicable to the motor as constructed. As a method of 
reviewing the problem, in a general way, however, the assump¬ 
tion made and the conclusions drawn therefrom are sufficiently 
exact. In the determination of the equations used above, an 
ideal motor has been considered, the resistance and local leakage 
reactance effects being neglected. Actual operating conditions 
may be more closely represented as follows: 


Let 

rf = resistance of field coil. 

r c = resistance of compensating coil. 

r a = resistance of armature. 

Xf = local leakage reactance of field coil. 

= local leakage reactance of compensating coil. 
x f = local leakage reactance of armature circuit. 

Then the copper loss of the motor circuits will be 
P Rm = Pr a + i 2 , (rt + r c ) = P [r 0 + 


( 202 ) 


where R m is the effective equivalent value of the motor-circuit 
resistance, that is, 


R 


m 


r a + 


rf + r c 
( pn 2 + p) 2 


(203) 


Similarly it may be shown that the equivalent effective value 
of the local leakage reactance of the motor-circuit is 


X 


m 


Xj + X c 

(. pn 2 + p ) 2 


(204) 


Combining equations (182), (203) and (204), the apparent 
impedance of the motor-circuits becomes 


Z = 




5 X t , r f + r c T 

n a (pit* T p) 2 ~j 


+ [X t + x a 4~ 


Xf + X c 


(: P n 2 + P ) 


j 


( 205 ) 










SERIES MOTORS. 


253 


The power factor is 


„ R 

cos 0= — = 


sx, +r 0+ «+* 


n 


(pn 2 + £) 2 


>l[ » +ra+ (M 2 +p)J. + [ x ‘ +ac<i+ (p^+J)J 

(206) 


£ 

The current is — — /. 

The input = E 1 cos 

The output is P = E I cos 6 — P R m . 


(207) 

(208) 
(209) 


£.|l*? +ra+ 


P 


D 


n 


(. pn 2 + p ) 


.] 


[ 


' ' ' a i / ^ „2 r >,\2 


W 


J+ [x,+*.+ 7 ^ ri 1 


(pn 2 + £) 2 J 1 |_. (j? » 2 + pj 2 J 

r ; + r J 


: 2 [r a 


+ 


{pn 2 + py 


[ 


SX/,.. , r/ + r c l 2 


n 


+ r a + 


+ |^X7 + x a -\- 


Xf + X c 


(pn 2 + p) 2 J 1 L“‘ W (pn 2 + p) 2 

E 2 SX t I 2 SXt 


J ( 210 ) 


P = 


Z 2 M 


n 


( 211 ) 


The torque is 

D P PX ‘ 


see eq. (187) and eq. (168) 


( 212 ) 


5 n 

Vector Diagram of Induction Series Motor. 

The graphical diagram of Fig. 117 represents the above im¬ 
pedance equations (e.m.f. for unit current), where 



OA -ra+.pF' 

(p n 2 +p) 2 

(213) 

A D 

__ Xf + X c 

— Xt +x a + {pn t + p) * 

(214) 


D F = —— at speed S 
n 

(215) 


OF = Z at speed 5 

(216) 

cos F O A 

= cos 6 = power factor at speed S 

(217) 




























254 


ALTERNATING CURRENT MOTORS. 


Although neglecting certain modifying effects, the graphical 
diagram represents quite closely the observed performance char¬ 
acteristics of the induction-series motor. An inspection of equa¬ 
tion (205) will show that certain values there given may be 
represented by others of much simplified nature since various 
terms there contained are constant in any chosen motor. 


Let, therefore, 


R=r + r, + Tc 
a (J>n 2 + p ) 2 

(218) 

v _ y , r , Xf + X c 

A — At + X a -\- 2 . , x2 

( pn 2 + py 

(219) 

*1* 

II 

Oh 

(220) 

then the apparent impedance becomes, 


2 = V(r+p sy+x 2 

(221) 

the power factor is, 


* R + PS 

cos 0 = —- 

V(r+p sy+x 2 

(222) 


which continually approaches unity with increase of speed. 


Generator Action of Induction Series Motor. 

Let rotation of the armature in the direction produced by the 
electrical (its own) torque be considered positive. Then may 
rotation in the contrary direction (against its own torque) be 
considered negative. Since the power component of the motor 
impedance has a certain value at zero speed, and increases with 
increase of speed, it should follow that by driving the rotor in a 
negative direction the apparent power component will reduce 
to zero and disappear. The power factor then reduces to zero 
and the current supplied to the motor will represent no energy 
flowing either to or from the motor. 

This will be apparent from the relations above set forth, as 
well as by the relations algebraically expressed by the equation 

power = E I cos 0 = ^ ~ ^ ^ . (22V\ 

X 2 + (R~P S ) {ZZ6) 

the negative sign being due to the direction of rotation and the 
expression reducing to zero for zero value of the apparent power 








SERIES MOTORS. 


255 


component, R — PS. A further increase of speed in the nega¬ 
tive direction will cause the expression for the power-factor and 
for the power, to become negative, the interpretation of which is 
that the machine is now being operated as a generator and 
hence is supplying energy to the line, that is, energy is flowing 
from the machine. Fig. 118 which gives the observed perform¬ 
ance characteristics of a certain induction-series motor, will 
serve to show to what extent these theoretical deductions may 
be realized in an actual machine. If, then, during operation as 



Fig. 118.—Test Characteristics of Induction-Series Motor. 

a motor at a certain speed, the quadrature field flux be relatively 
reversed with reference to the brush axial-line field flux, so as to 
tend to drive the armature in the opposite direction, not only 
will a braking effect be produced by such change but energy 
will be transmitted from the machine to the line. 

Brush Short-Circuiting Effect. 

The effect of the short circuit by the brush of a coil in which 
an active e.m.f. is generated, which has been omitted in the 












































256 


ALTERNATING CURRENT MOTORS. 


above equations, though completely included in the test curves, 
may be treated as follows. Referring to Fig. 117, it will be seen 
that at any speed 5 there will be generated in the coil under the 
brush by dynamo speed action an e.m.f. 

e s = K <j> a S see eq. (43) (224) 

where K is constant. This e.m.f. is in time-phase with the flux 
cj) a . In this coil there will also be generated an e.m.f., <?/, by 
the transformer action of the field flux, such that 

ef — K (f>f see eq. (44) (225) 

This e.m.f. is in time quadrature to cf>f. Since <£/ and (f> a are in 
time phase, the component e.m.f.’s acting in the coil under the 
brush are in time quadrature, so that the resultant e.m.f. is 


V e s 2 + ef — E 

V <j) a 2 S+<}>f' 

(226) 

is = n see eq. (175) 

9f 

(227) 

E b = K <j> a ^ 

s 2 +~ 

n 1 

(228) 


combining equations (169) and (181) 

2 7T / A/q ( f) a 

V2 ■ 10 s \| n 2 + 1 (229) 


4>a 


\/2 . 10 8 . £ 


2n)N a 1^1 +i 

\ n z 


combining (230) and (228) 


E h = 


K V2 . 10 s E 

2 71 f N a 


\l s,+ ? 


\ ( 


5 2 

~ + 1 

n 


(230) 


(231) 


^ „ IS 2 n 2 + 1 

A \l ( 232 ) 

where A is a constant as found above. 

When n = 1, E b is constant, independent of the speed, while 
when n is verv small E b is large at zero speed and continually 
















SERIES MOTORS. 


257 


decreases with increase of speed. When S = 1 or at synchronous 
speed 



K V2 10 8 E 
2~fN a 


(233) 


quite independent of the value of n. 

The relative impedance effect on E b can be determined by 
combining equations (232) and (185) thus 


E b _ A x t VS 2 n 2 + 1 7 _ 

I ~ En " (S 2 + w 2 ) ' VS2 + w2 


(234) 


E h 

Y = B \ / S 2 n 2 +1 (235) 

B being a constant. The interpretation of equation (235) is 
that the apparent impedance effect of the short circuit by the 
brush consists of two components in quadrature, one component 
being of constant value and the other varying directly with the 
speed. Experimental observations fully confirm these theoret¬ 
ical conclusions, and show that the increase in apparent reactive 
effect with increase of speed for motor operation is approxi¬ 
mately counterbalanced by the lagging counter e.m.f. (leading 
current) effect of the time-phase displacement between exciting 
current and field magnetism as has been mentioned previously 
and as will be dwelt upon subsequently. During generator 
operation, that is, with negative value of S, the apparent re¬ 
active effect of the short circuit at the brush adds directly to 
the lagging field flux, counter e.m.f. effect and therefore, the 
apparent reactance of the motor circuits increases rapidly with 
increase of speed in the negative direction, though remaining 
practically constant for all values of positive speed. These facts 
will be appreciated from a study of the test characteristics of 
the induction series machine throughout both its generator and 
motor operating range as shown in Fig. 118. 

Hysteretic Angle of Time-Phase Displacement. 

Mention has frequently been made of the fact that in the 
development of the equations for expressing the performance of 
the various types of series motors the effect of the hysteretic 
angle of time-phase displacement, between the magnetizing force 








258 


ALTERNATING CURRENT MOTORS. 


and the magnetism produced thereby has been neglected. In 
a closed magnet path operated at a density below saturation the 
tangent of the angle of time-phase displacement will be approxi¬ 
mately unity—depending for its exact value upon the quality 
of the magnetic material. Consider the magnetic and electric 
circuits of the machine treated as a stationary transformer. 
The hysteresis loss will be, in watts, 


Wh 


.0021 f A l B m l -° 
10 7 


(236) 


where A = cross sectional area of magnetic path 

l = length of magnetic path (in centimeters) 

B m = maximum magnetic density (c.g.s.) 

The electromotive force counter generated in the transformer 
coil having N turns will be, in effective volts, 


E = 


2 x f A B m N 
y/2 10 8 


(237) 


The current to supply the hysteresis loss will be 


Ih = 


Wh .0021 } A l 5 m 1 ' 6 vT 10 ! 
E ~ 2 x f A B m N ' 10 7 


= .000462 / B m ' 6 (238) 


With a permeability of p the magnetizing component of tne 
no-load current will be 


i,, - 


A B m l 


10 


B m l 


4- 

10 


p A \/ 2 Al 


4 V2 x ‘ l 1 N 


(239) 


For a certain value of permeability, depending upon the mag¬ 
netic density, the hysteresis current and the magnetizing cur¬ 
rent become equal in value. Thus when the two components 
of the no-load exciting current become equal Ip = Ih, 


V2 10 


.0021 l B m -« 
2 t N 


from which is obtained, 


B m .lA0 
4 7i \/2.P-N 


(240) 


p = 119 B m ' x 


(250) 


The meaning of equation (250) is that with a permeability of 
the value there designated, the hysteresis current and the no-load 













SERIES MOTORS. 


259 


e xciting cu rrent are equal in value and that the resultant current 
+ / /r is displaced from the flux by a time-phase angle whose 
tangent (equal at all times to the ratio of I jj. to Ik) is unity, as 
stated previously. For commercial laminated steel operated at 
densities below saturation, the permeability differs but slightly 
from the value given by the equation (250), though with increase 
of magnetic density above 7,000 lines per square centimeter the 
permeability falls off rapidly and the tangent of the angle of 
displacement between flux and current becomes correspondingly 
increased. * 

In an open magnetic circuit the permeability of a portion of 
the path reduces from the value approximately represented by 
the equation (250) to a value of unity, producing a very marked 
effect upon the hysteretic angle of displacement between flux 
and current. 


Let l = length of path in magnetic material of permeability jjl, 
d = length of path in air, 

then, assuming that permeability is as represented by equation 
(250), the tangent of the angle of time-phase displacement be¬ 
tween flux and magnetizing force is such that 


tan d = 


l_ 

V- 


l 


V- 


+ d 


l -t- ji d 


(251) 


the significance of which equation is that the flux lags behind 
the current producing it, by an angle which depends for its value 
largely upon the ratio of the air-gap to the length of the mag¬ 
netic path. Assigning values to /i, l and d, it will be seen that 
in any practical case the angle d must be quite small,—seldom 
more than 2 degrees. 

It should be carefully noted that a slight error is introduced 
on account of the fact that the permeability of commercial mag¬ 
netic material undergoes a cyclic change with each alternation 
of the current, and that, independent of the angle of time-phase 
displacement between flux and current, the shape of the waves 
representing the time-values of the two can not both be sinu¬ 
soidal, and that in assigning a value to the angle of time-phase 
displacement between the flux and current, the lack of similarity 
of the two waves has been neglected. 





260 ALTERNATING CURRENT MOTORS. 

Power Factor of Commutator Motors. 

Under speed conditions the e.m.f. counter generated by the 
cutting of the armature conductors across the field magnetism, 
varies in value with the magnetism, and hence it must have a 
wave shape of time-value similar in all respects to that of the 
field flux, and must have a time-phase position with reference 
to the field current quite the same as that of the magnetism. 
The counter generated speed e.m.f. must, therefore, lag behind 
the current by an angle whose tangent is as given by equation 
(251). Now since the counter e.m.f. lags behind the current, 
the current must lead the counter e.m.f. by the same angle— 
a fact which has been mentioned previously. 

With motors having air gaps of sizes demanded by mechanical 
clearance, the inherent angle of lead is quite small, and its effect 
upon the power factor is neutralized by the effect of the short 
circuit by the brush of a coil in which is generated an e.m.f. by 
both transformer and speed action when the machine is operated 
as a motor. When the machine is operated as a generator, how¬ 
ever, the hysteretic angle and the angle due to the short circuit¬ 
ing effect are in a direction such as to be additive to the station¬ 
ary reactive effect of the motor circuits and, therefore, during 
generator operation the power factor is lower than during motor 
operation, as shown in Fig. 118. 

While the angle of lead due to the hysteretic effect, even when 
the machine is running as a motor, is in any case quite small and 
its good effects cannot be availed of, it is possible by means of 
certain auxiliary circuits to give to the angle of time-phase dis¬ 
placement between the line current and the flux any value de¬ 
sired, and thus to cause the operating power factor to become 
unity or to decrease with leading wattless current, as is shown 
below. 

Resistance in Shunt w t ith Field Winding. 

Fig. 119 represents diagrammatically the circuits of a con- 
ductively compensated-series motor in parallel with the field 
coil of which is placed a non-inductive resistance. Consider 
first, ideal conditions in which the armature and compensating 
coils are without resistance and the compensation is complete 
so that these two circuits, trfeated as one, are without inductance. 
The field coil is without resistance but constitutes the reactive 
portion of the motor circuits. 


SERIES MOTORS. 


261 


When the armature is stationary the circuit through the re¬ 
sistance being open, the current taken by the machine has a 
value determined by the ratio of the impressed e.m.f. and the 
reactance of the field coil. This current lags 90 time degrees 



E s = Speed E.M.F. 



Fig. 119.—Circiut and Vector Diagrams of Compensated 
Series Motor with Shunted Field Coil. 

behind the e.m.f. across the field coils. When a resistance is 
placed in shunt to the field coil, current flows therethrough, 
quite independently of the field current. The current taken 
by the resistance is in time-phase with the e.m.f. impressed 
upon the field coil. 



























262 


ALTERNATING CURRENT MOTORS. 


In Fig. 119 let 0 I = If represent the field current, assumed 
always of unit value. 0 D = Ef is the e.m.f. impressed across 
the field coil and the shunted resistance. I r is the-current taken 
by the resistance. 0 C = /, the current which flows through 
the armature and compensating coil or the resultant current 
taken by the motor has a value represented by the equation 

I = v7/-+ i r 2 ( 2o2 ) 

and has a phase displacement ft with reference to the field cur¬ 
rent such that 

tan p=j- (253) 


With unit value of field current, under speed conditions, the 
e.m.f., E s , (D F of Fig. 119) counter generated at the brushes, 
due to the presence of the field flux, will be proportional directly 
to the speed and in time-phase with the field current. Thus 
this component of the counter e.m.f. of the motor is in no wise 
affected by the presence of the current through the shunted 
resistance. At a certain speed, the counter generated armature 
e m.f. will have a value represented by the line DF Fig. 119 

the resultant e.m.f. E = O F being the vector (quadrature) 

\ 

sum of the speed e.m.f. and the stationary e.m.f. E s , that is 

E = VEf + E? (254) 


and has a time-phase a position with reference 
E s such that 


tan a = 


Ff_ 

F s 


to the speed e.m.f. 


(255) 


An inspection of Fig. 119 will show 7 that under operating con¬ 
ditions, the angle of time-phase displacement between the cur¬ 
rent and the electromotive force, 6, has a value represented by 
the equation 

0 = ft — a (256) 


or the current leads the e.m.f. by the angle 6. At a certain 
critical speed for each value of shunted resistance, or at a certain 
value of resistance for any given speed, the angle 6 reduces to 
zero, and the power factor of the motor becomes unity. 

It is interesting to observe the effect of removing the resist¬ 
ance from in shunt with the field circuit. Since the current 





SERIES MOTORS. 


263 


taken by the resistance is 90 time-degrees from the field flux, 
the resultant torque due to the product of this component of 
the current and the flux is of zero value, the instantaneous 
torque alternating at double the circuit frequency. The cur¬ 
rent through the resistance, therefore, contributes in no way 
to the power of the machine or to the counter-generated, arma¬ 
ture-speed e.m.f., and when the circuit through the resistance 
is opened no effect whatsoever is produced upon the value of 
the current taken by the field coil, the counter e.m.f. or the torque 
of the machine. It is apparent, therefore, that the use of the 
shunted resistance increases the circuit current in a certain 
definite proportion, the added component being a leading 
“ wattless ” current under speed conditions. If a reactance be 
placed in parallel with the field coil, the current which flows 
therethrough will be in time-phase with the field flux, and the 
torque produced thereby will add to the torque due to the field 
current and it will affect directly the whole performance of the 
machine. The current taken by a condensance in shunt with 
the field coil will be in time-phase opposition to the field current 
and will tend to decrease directly both the circuit current and 
the armature torque. An excess of condensance will cause the 
torque to reverse and the machine to act as a generator even 
when the speed is in a positive direction. When the condensance 
and the field reactance are just equal, the circuit current re¬ 
duces to zero and the torque disappears. Under the conditions 
here assumed, the counter generated e.m.f. at the armature re¬ 
mains proportional to the product of the field flux and the speed, 
and there appears the remarkable combination of zero current 
being transmitted over a certain counter e.m.f. (that is, through 
infinite impedance) to divide into definite active currents at 
the end of the transmission circuits. 

Loss Due to Use of Shunted Resistance. 

From what has been demonstrated above, it is seen that 
shunted condensance acts to take current in phase opposition 
and to decrease the torque; reactance takes current directly in 
phase, and increases the torque, while resistance takes current 
in leading quadratures with the field current and has no effect 
upon the torque. It is evident that the improvement in power 
factor due to the use of the resistance is advantageous provided 


264 


ALTERNATING CURRENT MOTORS. 


the losses caused by the resistance are not excessive. Referring 
to Fig. 119, when the resistance is not used the power taken by 
the machine under speed conditions is 

P =01.0 F. cos F 0 I = If E cos a = If E s (257) 

When the machine is stationary, the power absorbed by the 
resistance is 

P r = C 1.0 D = I r E f (258) 

When the motor is running with shunted field coil, the power 
delivered to the machine is 

P t = OC.OF.cosCOF = I E cos 6 (259) 



Fig. 120.—Observed e.m.f.—Current Characteristics of Plain 
Series Motor with Shunted Field Coils. 


0 = P - a (260) 

cos 0 = cos j3 cos a + sin /? sin a (261) 

Pt — / cos /?. E cos a +1 sin /?. E sin a (262) 

Pt = If E s + I r E f = P + P r (263) 


The significance of equation (263) is that the power absorbed 
is that incident to the use of the resistance, and that for a given 
current it is unaffected by the speed e.m.f. Thus the current 
taken by the resistance multiplies into the stationary trans¬ 
former e.m.f. to give the actual watts absorbed while the same 

























SERIES MOTORS. 


265 


current multiplies into the speed e.m.f. to give apparent leading 
wattless power. 

In the derivation of the above equations ideal conditions have 
been assumed, which cannot be obtained in a practical motor. 
Fig. 120 represents the observed e.m.f.-current characteristics 
of a certain plain, uniform reluctance motor (see Fig. 114) 
with shunted field coils, and serves to show that even such an 
unfavorable machine may be caused to operate at unity power 
factor at any speed greater than about one-half synchronism. 


CHAPTER XV. 


PREVENTION OF SPARKING IN SINGLE-PHASE COMMUTATOR 

MOTORS. 

Transformer Action with Stationary Rotor. 

The greatest difficulty which has been encountered in the 
design of alternating-current motors of the commutator type 
has resided in the unavoidable e.m.f. produced at the coil under 
the brush due to the variation in the field magnetism. When 
the armature is stationary, there exists an appreciable electro¬ 
motive force between the terminals of each coil, the field coils 
acting as the primary and the armature coils in the neighbor¬ 
hood of the brushes as the secondary of a transformer, as indi¬ 
cated diagrammatically in Figs. 121 and 122. 

In motors of the repulsion type or by special magnetizing 
coils placed on any of the other types of motors, it is possible 
to neutralize the transformer e.m.f. in the coil under the brush 
by a speed generated e.m.f. when the rotor is in motion. See 
equations (48) and (101). The neutralization of the transformer 
e.m.f. under starting conditions is not wholly impossible, but 
it may be stated that such neutralization involves certain com¬ 
plications which are not desirable in a commercial motor. 

When the rotor is at rest the full effect of the transformer 
e.m.f. is felt at the brushes, quite independent of the type of 
motor employed and it may fairly be said that all simple forms 
of alternating-current commutator motors are equally dis¬ 
advantageous with regard to the sparking at starting. The 
current which flows through the short-circuit coil by way of 
the brush is ordinarily of large value, and it produces an exces¬ 
sive heating of the brush, the commutator segments and the 
coil. Moreover, the rupture of this current when the brush 
passes from one commutator segment to the next produces 
destructive arcing at the brushes, and its presence is in general 
detrimental to the perfect performance of the machine. To 
the evil effects of this local current in the short-circuited coils 

266 


SPARKING IN COMMUTATOR MOTORS. 


267 


may be attributed the slow progress which had been made in 
the development of the commutator type of alternating current 
machines previous to the last few years. 

Interlaced Armature Windings. 

The short circuiting effect may be largely eliminated by using 
two or more interlaced armature windings, so arranged that 
the brush cannot span the commutation sufficiently to connect 
two bars of the same winding. As usually applied, this method 
is not satisfactory on account of the fact that the current in 
each winding must be completely interrupted whenever the 
corresponding bar passes from contact with the brush. The 
interruptions occur at a frequency depending upon the speed 
of the rotor and the number of commutator segments, and they 
result in serious sparking and pitting at the commutator equally 
as disadvantageous as that caused by the short-circuiting. 
That is to say, the starting conditions have been slightly im¬ 
proved but the running conditions have become much worse. 

It has also been proposed to divide the armature circuits 
into three distinct interlaced windings, the current being led 
into the armature by way of two separately insulated brushes 
at each neutral point. By connecting the brushes in pairs to 
the terminals of two distinct secondaries of a single transformer, 
the current for the different armature windings shifts from one 
secondary coil to the other; but at no time is any armature or 
transformer circuit broken. The method here outlined renders 
the short-circuiting effect a minimum, and it possesses consider¬ 
able merit in this respect. However the method has not been ap¬ 
plied extensively in commercial practice, probably, on account of 
the involved electrical, and mechanical complications. 

Use of Series Resistance. 

Of the many methods which have been proposed for mini¬ 
mizing the effect of the short-circuited e.m.f. in the coil under 
commutation, those which involve the use of resistances in 
series with the coil have proven to be the most successful. 
Fig. 121 shows the method by which the resistances are inserted 
in circuit with the coil under the brush; the armature winding 
is closed on itself and is connected to the commutator through 
resistance leads. These leads serve the same function as the 


2G8, 


ALTERNATING CURRENT MOTORS. 


preventive coils used in alternating-current work when passing 
from one tap to another of a transformer. In fact this armature, 
in one sense, may be considered as a transformer with a lead 
brought out from each coil through a resistance to a contact 
piece, the various contact pieces being assembled together to 
form a commutator, as shown diagrammatically in Fig. 121. 

The function of the “ preventive ” resistance leads is to re¬ 
duce the short-circuit current, when passing from one bar to 
to the next to a desirable low value. As far as concerns com¬ 
mutation it is desirable that these resistances be as large as 
possible, while the loss of power due to the passage of the main 
motor current through them dictates that their value be kept 



Brush 


Resistance 

Leads 


Armature 

Coils 


Armature Lead 


Commutator 



Fig. 121.—Internal Preventive Resistance for Commutator Motor. 

quite small. It is evident, therefore that there is some 
intermediate condition which gives the most efficient results, 
both as regards the economy of power and the commutation 
of the current. 

Power Lost in Resistance Leads. 

It is worthy of note that although the prime object of the 
resistance leads is to diminish the short-circuit current and thus 
to minimize the sparking at the brushes, the losses are actually 
less when the resistance leads are used than when they are 
omitted. This fact will be appreciated when it is remembered 
that when the resistance leads are not used the loss due to the 
short-circuit current is enormous, although that due to the main 

















SPARKING IN COMMUTATOR MOTORS. 


269 


line current may be small. When resistance is inserted in the 
coil under the brush the former loss is decreased and the latter 
is increased. In practice the inserted resistance is given a 
value such that the sum of the two losses is a minimum, which 
condition exists when the two losses are equal. 

Internal Resistance Leads. 

The mechanical arrangement of the preventive resistances 
have not caused any very great difficulty in the construction 
of motors. Each lead is so placed that it forms a non-inductive 
path for both the short-circuit current and the main line current, 
which condition is conducive to sparkless commutation. Ac¬ 
cording to one method of construction, special slots are cut 
in the core for the reception of the leads. According to another 
method, the leads are placed in the same slots with the main 
armature winding. The resistance leads, after being insulated, 
are laid in the bottom of the slots, one terminal of each lead 
passing to a commutator segment and the other to a tapping 
point on the active armature winding which occupies the top 
portions of the slots. 

Objections which have been urged against the use of resist¬ 
ance leads relate to the power absorbed by the leads, and to 
the fact that, as ordinarily arranged on the armature, the re¬ 
sistance cannot conveniently be varied during the operation 
of the machine. Furthermore, although the leads are placed 
in positions where it is difficult to repair or replace them in 
case they are damaged, practical requirements demand that the 
leads be of limited cross-section, entailing the constant danger 
that they will burn out. Several schemes have been proposed 
for overcoming these objections. 

External Resistance Leads with Two Commutators. 

According to one of these schemes the motor is provided 
with two commutators connected to opposite ends of the arma¬ 
ture conductors, each commutator having alternate li\ e and 
dead segments. The brushes bearing on each commutator 
have a width not greater than that of a commutator segment. 
Several brushes of each polarity are distributed around each 
commutator, the brushes being so arranged that when one 
brush is on a dead segment other brushes of the same polarity 


270 ALTERNATING CURRENT MOTORS. 

are on live segments. The motor is arranged to be started 
with sufficient resistance between the parallel connected brushes 
to limit the short-circuit current to the desired amount, and 
this resistance is decreased as the motor comes up to speed. 

External Resistance Leads with One Commutator. 

Another scheme which accomplishes the same results with 
the use of only one commutator is indicated diagrammatically 
in Fig. 122. Instead of the usual single brush or set of brushes 
at each point of commutation there are employed three inde¬ 
pendently insulated brushes, each brush having a width some- 


Choke 

Coil 


u. 


Armature 


Lead 




Resistances 

K» 


X 


Brush 1 -! ii- 1 


Commutator 


Armature 

Coils 




Fig. 122. —External Preventive Resistance for Commutator Motor. 


what less than the width of a dead segment. The outer brushes 
are connected to the terminals of a reactance coil, while the 
middle brush is connected through resistances to the middle 
point of the reactance coil and to a terminal of the machine. 

It will be noted that when the outer brushes are on live seg¬ 
ments, the only current which can flow in the local circuit 
of the armature coil under the brushes and the reactance coil 
is the negligible exciting current of the coil. The main power 
current flowing through the armature passes differentially 
through the halves of the reactance coil and hence causes no 
opposing reactance. That is to say, for the line current the 
coil acts like a non-inductive resistance, but for the local short- 

























271 


SPARKING IN COMMUTATOR MOTORS. 

circuit current it acts like a true reactance coil to decrease the 
current to a negligible value. 

When the commutator moves to a position where the middle 
brush is immediately over a live segment, no current what¬ 
soever passes through the reactance coil, while the middle 
brush conveys the entire line current. In each of these two 
positions the machine is devoid of any short-circuiting effect 
and no abnormal heating is produced at any point. 

When the commmutator is in an intermediate position, 
however, where the middle and one outer brush are simultane¬ 
ously on the same live segment a disadvantageous short-circuit 
does exist. In the position here assumed an electromotive 
force having a value equal to one half of that of one armature 
coil tends to circulate a current locally through one-half of 
the reactance coil and the two brushes which bear on a single 
commutator segment. The adjustable resistances inserted 
in the circuit of the middle brush serve to keep the short-circuit 
current within proper limits. 

It will be noted that if the two outer brushes be placed on 
a single commutator segment, the reactance coil can be omitted 
and yet the resistances R 1 and R 2 may be employed to limit 
the value of the short-circuit current. In this latter event 
there are in effect only two brushes at each commutation point 
and there occur only one-half as many short-circuits per revo¬ 
lution as occur with the arrangement shown in Fig. 122. Sim¬ 
plicity would seem to dictate the use of two brushes without 
the reactance coil rather than three brushes with the coil. A 
little consideration will show that by inserting the reactance 
coil in circuit and dividing one brush into two parts the effect as 
far as commutation is concerned is exactly the same as though 
each segment of the commutator were live and the voltage be¬ 
tween segments were reduced to one-half of the value actually 
produced in each armature coil; that is, the reactance coil serves 
to obtain a middle e.m.f. point on each armature coil, and 
an armature provided with a one- turn- per- coil winding 
commutates as though there were only one-half turn per coil. 

In the arrangement shown in Fig. 122 the resistances are made 
of ample current carrying capacity for the maximum load on 
the machine, and they are, therefore, not subject to burn¬ 
outs. Moreover, they are external to the armature, and can 


272 


ALTERNATING CURRENT MOTORS. 


easily be adjusted or repaired. The reactance coils may be 
located at any convenient distance from the brushes of the 
machine and the connecting leads will serve as resistance to 
limit the value of the short-circuit current. Additional re¬ 
sistance can be inserted as found necessary, this latter resistance 
being varied at will during the operation of the machine. Thus 
the normal “ starting ” resistance may be employed simul¬ 
taneously to limit the short-circuit current at starting. 



APPENDIX. 


THE LEAKAGE REACTANCE OF INDUCTION MOTORS 

The Leakage Coefficient. 

As will be appreciated at once from a glance at the sim¬ 
plified circuit diagram and the circular current locus of an 
induction motor, the maximum power which the machine can 
possibly absorb at normal e.m.f. depends almost exclusively 
upon the combined local leakage reactance of the primary and 
secondary windings. It will be seen, therefore, that of all 
the calculations connected with the predetermination of the 
performance of a certain design none is of greater importance 
than that dealing with the leakage reactance. Although nu¬ 
merous theoretical equations are available for determining the 
reactance, the fact remains that only those that are formed 
upon an experimental basis are found to give results in con¬ 
formity to observation under test. In the outline treatment 
below an attempt is made to arrange in simple form certain 
design constants that have long been in use for predetermining 
the performance of induction motors, and to show the relation 
these constants bear to the leakage reactance, and thereby to 
investigate the facts upon which the reactance depends. 

Beyond any doubt the simplest and at the same time the most 
reliable equation that has ever been derived for assisting in 
the design of induction motors is the one given by Mr. B. A. 
Behrend on page S03 of the issue of the Electrical World and 
Engineer for November 24, 1900, namely, 

(1) 

in which a is the “ leakage coefficient,” as illustrated below; J is 
the radical depth of the air-gap; t is the pole pitch and C, which 


273 


274 


ALTERNATING CURRENT MOTORS. 


is designated as the “ dispersion factor,” depends for its value 
upon the arrangement of the slots and the conductors. Since o 
is a ratio, the value of C is independent of the units in which 
A and t are expressed. However, in what follows all lengths 
will be considered as measured in centimeters. 

The leakage coefficient, a , may be defined briefly as the ratio 
of the length, N 0 to 0 K , in the circle diagram of Fig. 54, 
reproduced herewith as Fig. 123. Referring now to the sim¬ 
plified circuit diagram of Fig. 48, shown here as Fig. 124, upon 
which the simple circle diagram is based, it will be noted that 
N 0 is the synchronous no-load exciting current of the motor, 
while 0 K has a value which would be represented by the 



Fig. 123.—Simple Circle Diagram of the Induction Motor. 


ratio of the primary voltage, E P , to the combined primary and 
secondary leakage reactance, (X P + X s ). Obviously when the 
value of o and either N O or O K are known, the main features 
of the circle diagram can be constructed and the maximum 
power factor and maximum input can be determined at once. 
On account of the convenience of the method, the value of 
N 0 is ordinarily first calculated from the mechanical dimen¬ 
sions of the rotor and stator, the arrangement of the primary 
coils and the magnetic density in the air-gap, and then 0 K 
is ascertained by the use of formula (1). It might thus appear 
that the “ short-circuit ” current is made to depend upon the 
radical depth of the air-gap, the magnetic density the pole 
pitch and the exciting current; as a matter of fact, however, 
the leakage reactance as found from the above equation, and 















LEAKAGE REACTANCE. 


275 


therefore the short-circuit ’ currents, are tacitly assumed to 
be entirely independent of all of these quantities. It is in¬ 
tended to give below a physical interpretation to the quantity, 
C, and to discuss in detail the facts upon which its value depends. 

Let the combined primary and secondary leakage reactance 
per phase be represented by Xp- \-x S} and let iq be the exciting 
current per phase and e the primary voltage per phase, then 


xp T x$ 

II 

(2) 

, a e 

^ A e 


x P + x s = — = 

Iq 

C ~7T 

t 'Iq 

(3) 


Fig. 124. — Simplified Circuits of Induction Motor. 



Exciting Current by Combined Magnetomotive Force 

Method. 

It remains now to determine the value of e in terms of the 
magnetic flux and the number of turns, and to ascertain the 
relation of i q to the magnetic density and the depth of the air 
gap. The methods of attacking these two problems may be 
divided into two general classes, one of which deals with the 
conditions existing when the secondary is on open circuit, and 
the other treats of conditions with the secondary short circuited 
while the rotor is at synchronous speed. To the lack of at¬ 
tempt to distinguish definitely between these two methods may 
be attributed the confusion and ambiguity which are found 
so frequently in discussions on this subject. The changes in 
the magnetic distribution and the value of the maximum 
magnetic density which are occasioned by opening and closing 



























276 


AL1ERNATING CURRENT MOTORS. 


the secondary when the rotor is at synchronous speed were 
treated at great length in Chapter IX, and they need not be 
dwelt upon here. It is well, however, to call attention to the 
fact that the equations for expressing the relation between 
the magnetic density and the primary e.m.f., and the corre¬ 
sponding equations for determining the exciting current from 
the maximum magnetic density vary enormously with the 
conditions of the primary and secondary circuits. See equa¬ 
tions (5), (6), (7), (10), (11), (14) and (15) of Chapter IX. 

It is possible to consider that the magnetism in the core is 
produced by the combined action of the currents of the several 
phases, and to derive both the primary e.m.f. and the primary 
exciting current on the basis of the maximum magnetic density. 
This is the method most commonly employed, the equations 
used by the various writers being based on the tacit assumption 
that the secondary circuit is open. According to this method 
it is necessary to determine the magnetomotive force required 
to produce the maximum magnetic density and then to assign 
■fro each phase its proper share of the exciting ampere-turns. 
The logical result of the complete application of this scheme is 
the (quadrature) exciting watts method treated at great length 
in Chapter IX. This latter method represents the extreme of 
the methods relating to the combined actions of the several 
phases when the secondary is either closed or open; it is very 
convenient for use when knowledge is had of the complete 
design data of the machine and it is necessary to determine 
accurately the value of the exciting current. 

Exciting Current by Single-Phase Method. 

For the purpose of the present discussion, it is desirable to 
call attention to another method which is sufficiently exact for 
preliminary design calculations. This method treats each 
phase winding just as though the other windings were not pres¬ 
ent, and it represents the extreme of the methods relating to 
the open secondary circuit condition. According to this method 
the flux threading each phase winding depends solely upon the 
e.m.f. impressed upon this winding, and the magnetic density 
is treated as being uniform over the area covered by each 
coil. Moreover, the magnetomotive force represented by the 
exciting current in each coil depends solely upon the value of 


LEAKAGE REACTANCE . 


277 


the uniform magnetic density just mentioned. It will be 
observed at once that if such a method as the latter can be 
applied, the magnetic calculations are rendered equally as 
simple as those relating to the magnetic circuits of a stationary 
transformer. That this method is correct for a two-phase 
motor when the secondary is open will be appreciated from a 
review of that portion of Chapter IX relating to Figs. 57 to 70, 
inclusive. It wall be observed at once that wffien one phase 
winding is acting alone the magnetic density is uniform over 
the coil area; wdien the other phase winding is active, although 
the interlinkage of lines in the first phase winding is the same 
as before, the magnetic density is no longer uniform. The 
exciting ampere turns in the first phase winding are not altered 
in value, how r ever, wdiile the exciting ampere turns of the added 
phase winding (which produces the non-uniformity of the mag¬ 
netic density) are exactly equal in value to the magnetomotive 
force w r hich w^ould be required if the added phase winding w'ere 
used alone 

It should be noted carefully that the above remarks are strictly 
applicable only under the conditions on wdiich Figs. 57 to 70 
have been based; that is to say, tw-o-phase windings with con¬ 
centrated turns, the air gap reluctance being uniform. It is 
sufficient here to state that the results obtained from applying 
the method to three-phase windings are sufficiently accurate 
for the present needs, while the modifications which must be 
introduced for absolute accuracy and to cover the cases where 
the coils are distributed along the air gap wffiose reluctance is 
in reality non-uniform w r ill be treated more fully below. 

Leakage Reactance Equations. 

Referring now- to equation (5) of Chapter IX, the e.m.f. per 
phase winding can be represented as 

c = / p w b t B m , ^4/ 

where f is the frequency, p is the number of poles, b is the width 
of the motor, n is the number of turns per pole per phase, b t 
equals the pole area and B m is the maximum instantaneous 
value of the average magnetic density in the air gap. 



278 


ALTERNATING CURREN1 MOTORS. 


Referring again to Chapter IX, and assuming that the re¬ 
luctance of the path in iron is negligible in comparison with 
that in air, but that the effective air-gap area is decreased by 
15 per cent., due the presence of the slot openings, equation 
(19) may be written 


11.5 B m A 


tq = 


4 7i a/2 


n 


Combining equations (3), (4) and (5), 


X* + Xs = C 


A y/2 nfnpbt B m 4 n V2 


n 


10 8 


11.5 B m J 


Thus, 


6.88 C , u 2 

Xp-\-x s — f pb n . 



( 6 ) 

(V) 


Considering momentarily that C is a true “ constant,” the in¬ 
terpretation of equation (7) would be that the effective leakage 
reactance per pole varies directly with the square of the number 
of conductors per pole, and as the width of the motor, and is 
independent of the pole pitch, the depth of air-gap, and the 
number of slots. 

A little consideration will show that the actual inter-linkage 
of leakage flux and conductors will vary somewhat with the 
number of slots in which the conductors are placed, that it is 
not entirely independent of the pole pitch, and that it will de¬ 
pend largely upon the reluctance of the leakage path; it will 
vary slightly with the depth of the air-gap, and largely with 
the shape of the slots. Upon these various modifying influences 
will depend the true value of the factor, C. 

In an article which appeared in the issue of the Electrical 
World and Engineer for April 30, 1904, Mr. H. M. Hobart re¬ 
ported the results of calculations and tests on 57 induction 
motors of eight different manufacturers in four different coun¬ 
tries, and gave curves, based on the tests of these machines, 
showing in what manner the leakage coefficient depends upon 
the pole pitch, the slot opening, the air-gap, and the number of 






LEAKAGE REACTANCE. 


279 


slots per pole. The above-mentioned curves have been repro¬ 
duced in Figs. 125 and 126 herewith, which, however, have been 
plotted with coordinates somewhat changed from the ones 
employed by Mr. Hobart, on account of the fact that the new 
scales chosen allow the results to be readily expressed in the 
form of simple equations that permit of easy interpretation. 
The “ dispersion factor/’ C, is considered as made up of two 
components, c and c'; the former depends upon the shape of 
the slots and the ratio of the pole pitch to the core length, and 



Fig. 125.—Main Dispersion Factor for Open and Closed Slots. 


the latter depends upon the depth of the air-gap and the number 
of slots, thus: 

C = cc'. (8) 


Values for c are given in Fig. 125, while Fig. 126 shows values 
for c '. 


Main Dispersion Factor. 


The heavy lines in Fig. 125 are the curves replotted from Mr. 
I-Iobart’s results. The curve for the completely closed slots 
can be represented with considerable accuracy by the following 
equation: 


Cr — 10.5 + 


2 9 t 



b 



























































280 ALTERNATING CURRENT MOTORS . 

Likewise the curve for the wide open slots is fairly well repre¬ 
sented by the equations: 

= 5.0 + (10) 

The graph of each of these equations is shown as a broken 
line near the corresponding curve. 

The interpretation of equations (9) and (10) w r hen used in 
conjunction with equation (7) would be that the leakage re¬ 
actance consists of two parts, one of which varies directly with 
the pole pitch and the other directly with the length of the core; 
for wide open slots, the latter is 1.72 times as large as the former 
per unit length, while with completely closed slots the latter is 
3.62 times as large as the former. 

It is permissible to assume that if all other conditions remain 
unaltered, the decrease in the leakage reactance for the “ em¬ 
bedded ” length varies directly with the percentage opening of 
the slots. Equations (9) and (10) can. therefore, be combined 
as follows: 


c 


10.5-5.5 S 0 + 


2.9/ 

b 


( 11 ) 


where S 0 is the percentage opening of the slots. Equation (11) 
gives the main dispersion factor. 

The true physical significance of equation (11) can be pointed 
out most clearly by assuming temporarily a^value of unity for 
the factor c' in equation (8), and combining equations (7), (8) 
and (11). Thus, 


x f + x s = (10.5 b — 5.5 S„ b + 2.9 t) f p w 2 , (12) 


which shows at a glance the relative importance of the “ em¬ 
bedded ” length and the “ free ” length of the primary coil in 
contributing to the local leakage reactance of the windings, as 
noted above. 




LEAKAGE REACTANCE. 


281 


Zig-zag Dispersion Factor. 

Values for c' as found by Mr. Hobart are shown by the heavy 
curve in Fig. 126. A fair approximation to this curve is given 
by the following equation: 


c' 


.o4 + 


.247 

Ah 


(13) 


w r here h is the average number of the stator and rotor slots 
per pole per phase. The broken line in Fig. 126 is the graph of 
equation (13). According to this equation a certain portion 
of the leakage reactance varies inversely w r ith the number of 
slots in which the conductors are placed, and also inversely 
with the depth of the air-gap. This portion may be designated 



as the “ zigzag ” leakage reactance, and c’ can be known as the 
“ zigzag ” dispersion factor. 

Combining equations (8), (11) and (13), the value of the total 
“ dispersion factor ” becomes 


C = ( 10.5 - 5.5 S 0 + AlCj (.54+ j-j) 


(14) 


It will be observed that the straight line graphs of equations 
(9) (10) and (13) do not coincide with the corresponding curves 
in Figs 125 and 126. It is worthy of note, however, that cal¬ 
culations of the leakage coefficients of the 57 differ** motors 
mentioned above when made according to the straight line graphs 










































































282 


ALTERNATING CURRENT MOTORS. 


agree with the actual test upon these machines equally as well 
as do those based on the heavy curves. Moreover, the equa¬ 
tions upon which the straight line graphs are based are inter¬ 
pretable strictly according to physical facts, while the inter¬ 
pretation of the deviations of the heavy curves from straight 
lines is involved in a considerable maze of guesswork. 

It is evident at once that the “ main dispersion factor ” con¬ 
siders all of the conductors of each pole winding to be cut by all 
of the leakage lines due to the currents in these conductors, while 
the “ zigzag leakage factor ” assumes the leakage lines due to 
the current in the conductors in each separate slot to surround 
only this particular slot. Neither of these assumptions is ab¬ 
solutely correct, but it is believed that the relative importance 
of the facts underlying the separate assumptions are properly 
accounted for in equation (14). 

Reactance of Coil-Wound Secondaries and Squirrel-Cage 

Motors. 

The values for the leakage reactance as given above are based 
on experimental observations made upon induction motors pro¬ 
vided with coil-wound secondaries. It is evident that the 
leakage flux in such motors varies with the mechanical position 
of the secondary coils with reference to the primary and is least 
when the rotor occupies such a position that each secondary 
phase winding is immediately opposite a primary phase wind¬ 
ing. A type of secondary in which the minimum leakage con¬ 
dition is found for almost all positions of the rotor is found in 
the “ squirrel-cage.” Experimental observations tend to show 
that the local leakage reactance of a squirrel-cage rotor is from 
60 to 80 per cent, of that of a similar motor with a coil-w r ound 
secondary. 

Taking the various factors into consideration and combining 
equations (7) and (14) the total leakage reactance per phase may 
be expressed as 

x = x P + x s = j^fpn 2 (10.56- 5.5S 0 5 + 2.9*). ^.54 + ^-^ ^ 

where K is a “ constant ” having a value varying from about 
6.25 to 7.50, a fair average value being 6.88, for induction 
motors having coil-wound secondaries; and having a value of 




LEAKAGE REACTANCE. 


283 


from 4.0 to 5.5, with an average of about 4.8, for “ squirrel-cage ” 
motors. 

For purpose of ready reference it is well to rearrange the 
various factors upon which the leakage reactance depends. Thus 
/ = frequency in cycles per second. 

b = width of motor (active length of slot) in cm. 

S 0 = slot opening (average of primary and secondary) with 
complete opening as unity. 

J = radial depth of air-gap in cm. 
p = number of poles. 
n = number of turns per pole per phase. 
t = circumferential throw of coil in cm. 
h = average number of the * rotor and the stator slots in 
which the coils of each phase are placed per pole. 

Reactance of Fractional Pitch Windings. 

It is desirable to explain the real significance of the quan¬ 
tities t and h, and to discuss the effect on the leakage reactance 
of using a fractional-pitch winding. When a full-pitch winding 
is used t is equal to the circumference at the air gap divided by 
the number of poles; but for a fractional-pitch winding t is equal 
to the “ pole pitch ” multiplied by the “ winding pitch-factor.” 
When a fractional-pitch two-layer winding is used, the coils 
of each phase are located in a greater number of slots than is 
represented by the average number of slots per pole per phase. 
The value to be given to h is the actual number of slots in which 
the pole windings of each phase are placed. Thus when there 
are four slots per pole per phase h is equal to 4 for a 100 per cent, 
winding pitch; but it may have any value up to 8 with a frac¬ 
tional pitch winding, according to the number of slots in which 
the coils of each phase are distributed—the fact that some of 
the slots may contain more turns of a certain phase winding 
than other slots may do, that is, that the turns per phase are 
not uniformly distributed in the slots, is of such minor im¬ 
portance that it need not be considered. 

Equivalent Single-Phase Reactance and Starting Cur¬ 
rent. 

Referring now to equation (15), for a two-phase motor with 
separate phase windings the equivalent single-phase leakage 


284 


ALTERNATING CURRENT MOTORS. 


reactance X P + X S will be one half of the value recorded in this 
equation, and the distance OK in Fig. 123 will be 2 E P + x. 

In a star-connected three-phase motor the equivalent single¬ 
phase leakage reactance will be exactly equal to the value given 
in equation (15) so that the distance 0 K in Fig. 123 will be 
E p -t~ x for a three-phase star-connected motor. 

In a delta-connected three-phase raotor the equivalent 
single-phase leakage reactance will be two-thirds of the value 
indicated in equation (15). Thus the distance OK in Fig. 123 
will be 3E P + 2x for a delta-connected three-phase motor. 

Calculation of Synchronous No-Load Currents. 

Referring now to the circuit diagram of Fig. 124, it is to be 
noted that the core loss current can be calculated Qnly when 
knowledge is had of the flux density in each magnetic path of 
the core. The value of this density can be determined most 
readily by means of equation (14) or (15) of Chapter IX. The 
true value of the “ exciting current ” can be ascertained by the 
use of equation (23) of the same chapter, the quantities of 
which are identical with those required for calculating the core 
loss current from equation (14) or (15). The results from the 
use of these equations will in general be more satisfactory than 
can be obtained from equation (5) given above. The latter 
equation is based on certain approximations which correspond 
with the conditions stated, and it is not applicable directly to 
conditions differing greatly therefrom. It must not be in¬ 
ferred, however, that the reactance equations given above are 
thus limited in application because the wide range in the “ con¬ 
stant” C is sufficient to cover all practical operating conditions. 

It should be noted carefully in this connection that all of the 
equations here recorded deal with empirical constants, and it 
is not permissible to extrapolate beyond the range covered by 
the tests. It is believed, however, that for induction motors 
of relatively standard pattern, excluding all freak designs, the 
equations and curves furnish a reliable basis for calculations. 


INDEX 


Air Gap, effect of volume on-excit¬ 
ing current of induction mo¬ 
tors, 135, 275, 284. 

Alternators (see Generators). 

Angle of lag, determination of, 11. 
determination of with one 
watt-meter, 6. 

Arc lamps, frequency required by, 
33. 

Armature turns effective, 216. 
windings interlaced, 267. 

Auto starter for induction motors, 

22 . 

Behrend, leakage coefficient, 275. 

Circle diagram, 65, 76, 100, 106, 
109. 

accuracy of, 106, 108. 
errors in, 71, 99, 107. 

Circuits, electric and magnetic, 95. 
equivalent electric, 98, 114, 273. 
magnetic reluctance, effect of 
varying, 96, 145. 
single-phase and polyphase, 1. 

Coils, effect of grouping on rating 
of induction motors, 133, 
283. 

in series, effective value of e.m.f. 
in, 131, 150. 

Commutating motors (see motors). 

Commutator on rotor of single¬ 
phase induction motor, 56, 
191, 201. 

used to excite asynchronous gen¬ 
erators, 92, 189, 201. 


Concatenation control, 23. 

Condensance, adjustment of, 89. 
used as source o: exciting cur¬ 
rent, 83. 

use in split-phase motor, 62. 

Condensers, operation of, 84. 

Condenser, synchronous, 81. 

Conducting material, economy of, 1. 
required for different transmis¬ 
sion systems, 3. 

Control of induction motor by con¬ 
catenation, 23. 

Converters, frequency, 33. 
capacity of, 35. 
field of application, 33. 
performance of, 34. 
power supplied by, 35 
inverted, 170. 
motor, 36. 

advantages, 38. 
excitation, 39. 
phases, 40 
uses, 40. 
six-phase, 174. 
synchronous, 149. 36, 
capacity, relative 154. 

for various phases, 163. 
characteristics, of 172. 
compounding of, 169. 
current (a.c.) maximum, value 
of, 155. 

equations of, 173. 
excitation of, 166. 
heat loss in armature coils, dis¬ 
tribution of, 159. 


285 




286 


INDEX. 


Converters, frequency, continued. 
hunting of, 167. 
operation at fractional power 
factor, 160. 

operation at unity power fac¬ 
tor, 154. 

performance, characteristic, 
165. 

performance, predetermina¬ 
tion of, 171. 
starting of, 168. 
two-phase, currents in, 156. 

Core> effect of volume on exciting 
current of induction motors, 
135, 284. 

flux in induction motors, deter¬ 
mination of 130, 275. 

Currents, equivalent single-phase, 
13, 15, 283. 

Delta vs. star-connected -Drimaries, 
180. 

Diagram of compensated repulsion 
motor, 221. 

of compensated series motor. 

with shunted field coil, 2b 1. 
of induction series motors, 246. 
of inductively compensated se¬ 
ries motors, 240, 241. 
of performance of polyphase in- 
‘ duction motors, 65, 76, 100, 
108, 109. 

of performance of single-phase 
induction motors, 115, 193, 
207. 

of plain series motor, 233. 
of repulsion motor, 193-206. 

Dispersion factor, 279, 281. 

Double current generators (see gen¬ 
erators). 

Eddy current losses, 98, 188. 

Effective, armature turns, 216. 

Electrical-space degrees, 123. 

Electrical-time degrees, 123. 

E.m.fs., generated by an alternat¬ 
ing field, 121, 123, 131, 197, 
199. 


E.m.f.’s continued. 

in group of coils in series, effec* 
tive value of, 133, 150. 
in six-phase circuit, 2. 
in three-phase circuit, 2. 
Electromagnetic torque, 181. 
Equivalent circuits, 98, 114, 273. 
single-phase currents, 13, 283. 
single-phase resistance, 14. 
single-phase quantities, 15. 
Excitation of induction motors, 42, 
73, 135, 284. 

of synchronous machines, 166. 
Exciting watts in induction motors, 
73, 135, 284. 

Ferraris’ method of treating single¬ 
phase induction motors, 138. 
Fisher-Hinnen device for starting 
induction motors, 23. 
Four-phase, 156. 

Frequency converters (see Convert¬ 
ers). 

effect upon alternators in paral¬ 
lel, 33. 

required by arc lamps, 33. 
used in rotary converters, 33, 165. 

Generators, asynchronous, 74. 
characteristic performance, 79. 
core, design of, 91. 
commutator excitation of, 92. 
compensation for inductive load, 
94. 

connections for condenser exci¬ 
tation, 89. 

current diagram, 75. 
excitation, characteristics of, 86. 
excitation of, 81. 
excited by condensance, 83. 
lagging current load, effect of, 90. 
leading current load, 90. 
load characteristics of, 90. 
operation of, 75. 

operation with commutator ex¬ 
citation, 93. 

parallel operation of, 80. 
performance, calculation of, 77 



INDEX. 


287 


Generators, synchronous, continued 
shunt excited, 94. 
d. c., capacity compared with va¬ 
rious a. c. machines, 164. 
double current, 149, 161. 

capacity for various phases, 
163. 

heat loss in armature coils, 
distribution of, 159. 
synchronous, 151. 

capacit/ compared with d.c. 
machines, 152. 

capacity for various phases, 
163. 

parallel operation of, 80. 

Heyland asynchronous generator, 
94. 

induction motor (see Motors). 

Hobart test results, 278. 

Hunting, 167. 

Hysteresis losses, 98. 

Induction motor (see Motor, Induc¬ 
tion). 

Induction series motors (see Mo¬ 
tors, Series Induction). 

Inverted converters(seeConverters) 

Leakage reactance in compensated 
repulsion motors, 229. 
coefficient of, 275. 
in induction motors, 73, 273. 
calculation of, 277. 
complete discussion of 273. 
in induction series motors, 252. 
in repulsion motors, 209. 

Loading back method of measuring 
torque, 185. 

Loads balanced, 6. 
unbalanced, 9. 

Magnetic distribution in polyphase 
motors, 125, 131, 284. 
dispersion 279, 281. 

Measurements of power, 3. 

Mechanical-space degrees, 123. 


Methods, graphical, advantage of 

63. 

Motor-converters (see converters 
motor). 

Motors, commutator, e.m.fs., gener¬ 
ated by alternating field, 199 
power factor, 260. 
power lost in resistance leads 
268. 

sparking, prevention of, 266. 
torque, production of, 181. 
treatment of, simplified, 189. 
induction, alternating current 
in secondary coils, 42. 
capacity, method of increasing, 
119. 

concatenation control, 23. 
core flux as affected by dis¬ 
tributed winding, 131, 283. 
core flux, determination of, 130 
current diagram at all speeds, 
75. 

locus, 65, 106, 108, 109, 116. 
locus, effect of resistance on, 
108. 

equation, 70, 101. 
exact, 107. 

design, effect of leakage react¬ 
ance on, 73, 282. 
direct current in secondary 
coils, 42. 

efficiency, 21, 32, 69, 110. 
equations, analytic, 19, 26, 

64, 70, 130. 
excitation of, 41. 

exciting watts, value of, 73, 
135, 284. 

Fisher-Hinnen device, 23. 
general outline, 16. 

Heyland, 42. 

action with rotor at syn¬ 
chronous speed, 46. 
action with stationary rotor, 
44. 

connecting resistance, func- • 
tion of, 43. 

direct current armature cf 
43. 




288 


INDEX. 


Motors, induction, continued. 

internal voltage diagram, 105. 
iron losses, 98. 284. 
magnetic field in, 121. 
method of treatment, 16. 
operation above synchronism, 
75. 

below synchronism, 74. 
outline of characteristic feat¬ 
ures, 48. 
output, 21. 

performance observed, 25. 
performance diagram, proof of, 
111 . 

polyphase, capacity of, 119. 
capacity as affected by grou¬ 
ping of coils, 133. 
compared with single-phase 
motors, 112. 

exciting current, 73, 135. 
exciting watts, 73, 135. 
magnetic distribution with 
closed secondary 127. 
magnetic distribution with 
open secondary, 124. 
magnetic field in, 121. 
output, maximum, 110. 
operated as single-phase ma¬ 
chines, 58. 

performance, complete dia¬ 
gram of, 109. 

power factor, maximum, 110. 
torque, maximum, 110. 

power factor, 21. 

calculation of, 29, 66, 110. 
method of improving, 41. 

primary current, calculation of 
30, 284. 

reactance at standstill, 18, 73, 
273. 

resistance external to second¬ 
ary windings, 23. 
in secondary windings, 22. 
revolving field, production of, 
17, 125. 

secondary current, determina¬ 
tion of, 28, 66, 116. 


Motors, continued. 

effective resistance in, 63, 
116. 

exciting m.m.f., 41, 277. 
frequency, 18. 

resistance measurement of, 
28, 116. 

series commutator, 245. 
single-phase, 48, 112, 138. 
capacity of, 119, 134. 
commutator for starting 
purposes, 56, 189. 
compared with polyphase, 
48, 112, 138. 

current in secondary, 49, 
1436. 

electric circuits of, 114. 
equivalent circuit (exact), 

114. 

equivalent circuit (modified) 

115. 

magnetic field in, 138, 140. 
magnetizing current, 54, 143 
performance, diagram, 115, 
193, 207. 

polyphase motor, used as, 
58, 119, 134, 138. 
quadrature magnetism, pro¬ 
duction of, 49, 138. 
revolving field, elliptical, 

53, 146, 194, 213. 
circular, 52, 146, 194. 
production of, 51, 138, 

194, 213. 

secondary currents in, 1436. 
secondary quantities, graph¬ 
ical representation of, 146. 
shading coils, 54. 
speed equation, 117. 
speed field, 117, 139. 

as affected by speed, 148. 
current production of, 139, 
143. 

torque, 117. 

action of commutator in 
producing, 56, 189, 201. 
starting, 53, 189, 201. 
transformer features of, 142. 



INDEX . 


289 


Motors, continued. 

transformer field, as affected 
by speed, 148. 

slip measurement of 26, 110, 
117. 

starting devices for, 22, 53, 189 

201 . 

synchronous speed, determina¬ 
tion of, 24, 194, 204, 213, 
230. 

method of decreasing, 24. 
tandem control, 24. 
test with one voltmeter and 
one wattmeter, 25. 
three-phase, equivalent start¬ 
ing current, 113, 283. 
starting current (equivalent 
single-phase), 113, 283. 
operated on single-phase cir¬ 
cuit, 61, 119, 134, 138. 
torque, determination of, 27, 
110, 184. 

maximum, 20, 69, 110. 
transformer features of 95,121. 
treatment of, graphical, 63. 
106, 109. 

two-phase, equivalent starting 
current, 113, 284. 
operated on single-phase cir¬ 
cuit, 59, 119, 134. 
starting current (equivalent 
single-phase), 113, 284. 
test results, 29, 66, 76. 
used as frequency converters, 
31, 36, 41. 
generators, 74. 
motor-converter, 36. 
synchronous motor, 39. 
voltage, internal, 105, 114, 

121, 142. 

winding distributed, 131, 283. 
effect of, on core-flux, 131, 
283. 

repulsion, 56, 189. 

brush, short-circuiting effect, 

212 . 

characteristics of, 197, 198, 

214 . 


Motors, repulsion, continued. 
compensated, 214. 

brush, short-circuiting effect 
228. 

characteristics of, 222. 

fundamental equations of, 
218. 

leakage reactance, 227. 
performance, calculation of, 
223. 

observed, 224. 
resistance, 227. 
test, 225-226. 
vector diagram of, 221. 
construction of, 201. 
diagram proof of, 208. 
electrical circuits (ideal), 190. 
equations of, 210. 
fundamental equations of, 204. 
graphical diagram of, 193. 
graphically treated, 199. 
impedance apparent, 217. 
leakage reactance, 209. 
magnetic circuits (ideal), 190. 
operation of, 202. 
performance, calculation of, 
195. 

observed, 213. 
resistance of, 209. 
test of, 214. 
torque, 209. 

production of, 192. 
treatment, algebraic, 199. 
vector diagram of, 206. 
series, 232. 

compensated conductively, 242 
equations, 243. 
impedance, 242. 
inductively, 241. 
line current, 243. 
performance, calculation of, 
243. 

power, 243. 
power factor, 242 
torque, 243. 
vector diagram, 244. 
field winding shunted with re- 
s'stance, 260. 



290 


INDEX. 


Motors, series, continued. 

fundamental equations with 
non-uniform reluctance, 237. 
with uniform air-gap reluc¬ 
tance, 233. 

hysteresis loss in, 258. 
induction, 245. 

brush, short-circuiting effect 
255. 

equations, fundamental, 247 
generator action in, 254. 
leakage reactance, 252. 
resistance, 252. 
starting torque, 250. 
test of, 255. 
vector diagram, 253. 
hysteretic angle of time-phase 
displacement, 257. 
loss in shunted resistance, 263. 
performance with shunted field 
coils, 264. 
power, 236. 
power factor, 235. 
resistance in shunt with field 
winding, 260. 
torque, 236. 

treatment, algebraic, 232. 
graphical, 232, 
synchronous, 149, 151. 
capacities relative, 154. 
excitation of, 166. 
hunting of, 167. 

Performance diagram for polyphase 
induction motors, 65, 76, 
106, 109. 

of single-phase induction mo¬ 
tors, 115. 

of simple repulsion motor, 
193, 207. 

of compensated repulsion mo¬ 
tor 221, 222. 

of plain series motor, 233. 
of compensated series motor, 
240, 241, 261. 

of induction series motor, 243. 

Phase relation of voltages in syn¬ 
chronous machines, 150. 


Polyphase circuits, 1. 

induction motors (see Motors, 
Induction). 

Power, apparent in three-phase cir¬ 
cuit, 13. 

factor, adjusted by resistance in 
shunt with field winding, 260. 
determination of, 11, 29, 66, 
109. 

maximum of induction motors, 
36, 110. 

method of improving, 41. 
measurements, three-phase, 3. 
unbalanced three-phase cir¬ 
cuits, 3. 

three-phase, one wattmeter 
method, 6. 

two wattmeter method, proof 
of, 3. 

Quadrature watts, 8, 135, 284. 

Reactance, leakage, 73, 273. 

in compensated repulsion mo¬ 
tors, 229. 

in induction series motors, 252. 
in repulsion motors, 209. 
of induction motors running, 
18, 73, 273. 

of induction motors at stand¬ 
still, 18, 73, 273. 

Resistance in secondary winding of 
induction motor, 22. 
equivalent, single-phase, 14. 
in shunt with field winding, 260. 
used to prevent sparking, 267. 

Rotary converters (see Converters). 

Revolving field, production of, 17 
51, 112, 121, 125, 143, 194, 
213, 231. 

Ryan balancing coils, 94. 

Scott transformer connections, 177. 

Shading coils, 54. 
action of, 54. 

Single-phase circuits, 1. 

induction motors (see Motors, 
Induction). 



INDEX. 


291 


Six-phase transformation, 175. 

Slip, 18, 26, 110, 117. 
measurement of, 26, 110, 117. 

Sparking in commutator motors, 
212, 228, 255, 266. 
power lost in resistance leads, 268 
prevention of, by external resist¬ 
ance leads with one commu¬ 
tator, 270. 

by external resistance leads 
with two commutators, 269. 
by internal resistance leads, 
269. 

by interlaced windings, 267. 
by series resistance, 267. 
transformer action with station¬ 
ary rotor, 28, 44, 63, 142. 
198, 266. 

Split phase motor, 58. 

use of condensance with, 62. 

Star vs. delta-connected primaries, 
180. 

Starting devices for induction mo¬ 
tors, 22, 53, 191, 203. 

Steinmetz hysteresis formula, 97. 

Synchronous commutating ma¬ 
chines, definition of, 149. 
converters see (Converters), 
motors (see Motors.) 
speed, determination of, 24. 


Three-phase to six-phase transform¬ 
ation, 175. 

lorque, determination of, 27, 110, 
184. 

electromagnetic, 181. 
for non-uniform reluctance, 183. 
for uniform reluctance, 181. 
measurement of, 185. 

electrical, errors in, 187. 
production of in commutating 
motors, 181. 

Transformer action with stationary 
rotor, 28, 44, 63, 142, 198, 
266. 

circle diagram for, 100. 
connections, 175. 
equivalent circuits of, 99. 
equivalent circuits of (approxi¬ 
mate), 99. 

hysteresis loss in, 97. 
iron losses, 98. 
principle of, 95 
six-phase, 175. 

used to adjust condensance, 89. 

Two-phase to six-phase transform¬ 
ation, 175. 

Windings, armature, interlaced, 267 
distributed, 131, 283. 


Tandem control of induction mo 
tors, 24. 


Zigzag leakage, 281. 





































